Fractional Quantum Hall Effect

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Fractional Quantum Hall Effect

Dear Friends:

As I said last week I have started developing a paper that will incorporate the review I received from PRD for the Yang-Mills monopole paper, and fundamentally, the thesis will migrate over to "Why SU(3)_C t’Hooft Polyakov Monopoles are baryons." The ultimate results will be the same; but the connections to the scalar fields will be much more explicitly developed.

The other night I was getting a start on the first section of this paper, which was intended to be a "review" of Dirac Monopoles and the Dirac Quantization Condition without strings, using the gauge field method pioneered by Wu and Yang, which Zee reviews very nicely at https://jayryablon.files.wordpress.com/ ... opoles.pdf. This was to serve as an introduction to t'Hoot-Polyakov monopoles, and to rebut point 3 of the PRD review of the classical side of the theory, which as I shared last week was the following:
PRD Chief Editor wrote:3) The surface integral of F is not gauge-invariant, and neither are the other surface integrals of quantities that you write down. By appropriate local gauge transformation any nonzero surface integral of F can be made to vanish. This not only invalidates your Eq. (3.3), but shows that physical conclusions cannot be drawn from the value of this quantity.

In the process of doing this development, it occurred to me that certain solutions of the Dirac monopole had never been fully developed, and as I explored those states which had been "left on the table" by others, I began to realize that this could be used to account fully and completely for the fractionalized charge pattern observed in the Fractional Quantum Hall Effect. So over the weekend I wrote up an 11 page paper you may review at http://vixra.org/pdf/1411.0552v1.pdf, which as of Thursday I had no anticipation of writing. I do not think of myself as a condensed matter theorist, so I felt a bit like a fish out of water, but managed to do enough review of this area and got some private comments from a friend who is a professional condensed matter theorist, all of which made me comfortable I was not too badly off any bases. I also submitted this paper to PRD, and will let you know what happens.

Here is the Abstract:
The purpose of this paper is to explain the pattern of fill factors observed in the Fractional Quantum Hall Effect (FQHE), which appears to be restricted to odd-integer denominators as well as the sole even-integer denominator of 2. The method is to use the mathematics of gauge theory to develop Dirac monopoles without strings as originally taught by Wu and Yang, while accounting for orientation / entanglement relationships between spinors and their environment in the physical space of spacetime. We find that the odd-integer denominators are included and the even-integer denominators are excluded if we regard two fermions as equivalent only if both their orientation and their entanglement are the same, i.e., only if they are separated by 4π not 2π. We also find that the even integer denominator of 2 is permitted because unit charges can pair into boson states which do not have the same entanglement considerations as fermions, and that all other even-integer denominators are excluded because only integer charges, and not fractional charges, can be so-paired. We conclude that the observed FQHE fill factor pattern can be fundamentally explained using nothing other than the mathematics of gauge theory in view of how orientation / entanglement applies to fermions but not to bosons, while restricting all but unfractionalized fermions from pairing into bosons.

Best to all,

Jay
Yablon
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Re: Fractional Quantum Hall Effect

To all:

I have spent a lot of time thinking about my most recent findings about FQHE at http://vixra.org/pdf/1411.0552v1.pdf that the number of "windings" of a charge determines its fractional charge. As the paper shows, orientation and entanglement need to both be maintained, and then one can get to the n/3, n/5, n/7, n/9... fill factors with odd integer denominators. The pertinent orientation and entanglement material from Misner, Thorne and Wheeler (MTW) is linked at http://www.math.utah.edu/~palais/mtw.pdf. All of this came from mathematics; now I want to understand the physics. Or, from MTW: "Whether there is a detectable difference in physics ... for two independent versions of an object is not known." I want to know this. Attention Joy: I think this connects to what you are doing also.

So I spent a good part of the day asking myself how the "environment" of a charge would "know" that it has been rotated by two turns, or four turns, or six turns, etc. once it had been "untangled," because that is what causes the observed fractional charge. I also was asking what it really means to "rotate" a charge at all for orientation / entanglement purposes, because electrons are spinning all the time so one could say that every spin cycle is a turn. But I also kept in mind that and electron is a stationary object, and that the effects of its spin are captured in the magnetic filed not the electric field. So even with spin, the electron, and its electric and magnetic fields, have a stationary configuration. So if we want to talk about orientation and entanglement, we should be talking about the field lines, not about the electron.

It next occurred to me, channeling Faraday, that the electric field lines can be thought of as the "threads" which in MTW's http://www.math.utah.edu/~palais/mtw.pdf connect the "ball" to its environment for entanglement purposes. And "rotating" the electron for entanglement purposes would really mean rotating the places where the electric field lines emanate out from the electron. That is, look at any set of electric field lines such as http://upload.wikimedia.org/wikipedia/c ... zontal.svg and think of them as the threads in MTW's http://www.math.utah.edu/~palais/mtw.pdf connecting the electron to its environment. Then do a rotation through two turns. The field lines will themselves get tangled, and then they can be untangled. There is a good animation that you should see which shows orientation / entanglement in its simplest form, at http://commons.wikimedia.org/wiki/File: ... lement.gif. So from all of this, this is what I was thinking:

After a two turn = 4pi rotation, yes, the orientation is restored and the threads can all be untangled. But there is something that is still missing in all of this analysis: The threads themselves will get twisted, and if the threads are twisted and are stand-ins for electric field lines, then back to Farday: the field lines are twisted!!! And that would give you a "record" in the field that even though the fields met the electron in the same place and the fields had all been untangled, the field lines themselves in their twists would store information about how many turns had been made and this would be something physical.

So I did an experiment based on http://commons.wikimedia.org/wiki/File: ... lement.gif. I do not have a fancy lab, but I do have two pencils and two rubber bands. I cut the rubber bands, marked one side of each rubber band in dark marker, tied them to the pencil, and did the orientation / entanglement process shown in http://commons.wikimedia.org/wiki/File: ... lement.gif to see what happened to the field lines (rubber bands) themselves. And when doing so, I took five photos which I have posted in a five page file at https://jayryablon.files.wordpress.com/ ... -twist.pdf. Page 1 shows the starting configuration. Page 2 is one turn. Page 3 is two turns. Page 4 is the first untangling move. Page 5 is what you get after the complete untangling.

Study page 5: Yes, following two turns and untangling, the orientation is restored and the entanglement is restored. But the field lines are twisted. Following two turns, each rubber band has two twists, with opposite parity: the band on the right is right handed, the band on the left is left-handed. Spin up and spin down. The total number of twists on the left is -2 and on the right is +2, totaling zero, but the twists are still there. If you do four turns then untangle, you get four twists in each rubber band. And so on.

Now, I do not know if I am the first person to notice that if you do the orientation / entanglement drill,the threads themselves pick up a twist even after they are untangled. My guess is that I am not. But I do know that this gives me the physical explanation for the fractional charge in FQHE which my new paper http://vixra.org/pdf/1411.0552v1.pdf reveals mathematically: If I have rotated the electric field lines emanating from an electron over two turns (note I said "rotate the electric field lines," not "rotate the electron"), then after untangling, every field line will contain two twists. Some left handed,some right handed, all netting out to zero, but still two twists. This is the fractional charge of 1/3. Four turns and untangle, then four twists and 1/5 charge. Six turns and untangle, then six twists and 1/7 charge. The environment knows what has happened even after the untangling. The evidence remains, and that affects the charge as it is physically observed.

Apparently, physically, when a field line get twisted like this, that diminishes its strength. The electron is still the same physical electron, but it is emitting field lines which have their own twists. Mobius has come home to roost, and we are into full bore topology. When a field line has 2 twists, 2/3 of its charge is suppressed, leaving 1/3 (and unit multiples of 1/3) observed in FQHE. With 4 twists: 4/5 of the charge is suppressed, 1/5 or quantized multiples is observed. With 6 twists 6/7 is suppressed. So all of this quasi particle business, at least regarding the electron itself, is the result of the electric field lines of the electron becoming twisted and losing a t/(t+1) portion of their observed strength for t twists.

Now AFAIK, all of this only happens in a superconducting material and at very low temperature. So what this means is that in this environment, the electric fields are able in some fashion to "loosen up" from the electron such that they can undergo a rotation that tangles them up relative to their environment, then they can become untangled, but the field lines keep the twists and this then fractionally reduces the observed charge. Put perhaps more formally, in ordinary environments, the field lines of an electron are stationary. But in a near-Kelvin superconducting environment, the field lines are no longer stationary, they can rotate at the source, they can get tangled and untangled, but when they are untangled, they will retain information in their twists which does fractionally affect the magnitude of the observed charge. I think of this visually, in Faraday mode, as field lines "floating" over the surface where they connect to the electron.

I will write this up more formally when I have a chance, but I wanted to share this right away. I think this also connects very much to what Joy Christian has been doing with quantum correlations, because he is tying two electrons to their environment through entanglement also, and then separating those electrons and asking what happens next. My answer to MTWs question ""Whether there is a detectable difference in physics ... for two independent versions of an object is not known." Yes, there is. It is the FQHE. Field lines can have twist even if they are not tangled, and when they do, the strength is diminished by t(t+1) for t twists. Add this twisting field line behavior to what Faraday first taught us about fields.

Jay
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Re: Fractional Quantum Hall Effect

Wow! I had never heard of this before. Perhaps this is something new.
FrediFizzx
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Re: Fractional Quantum Hall Effect

Hi Jay,

Congratulations again! My brain has today come out of an influenza haze and I have now read your paper (as best I can) and compared your ideas with my preon viewpoint. There is a lot in it that gives me more questions.

The elastic band experiments: I can believe your findings are true. If you attach an elastic string from a fixed anchor point A to the end (B) of a pencil and then twist the pencil n times then there must be n twists of the elastic. So any rotation always leaves the elastic in a different twist state than before, irrespective of any re-jiggling of the pencil to free the elastic which does not involve twisting the pencil. So a 4π rotation of an electron does not get it back to its original state? The words microscopic and macroscopic are not correct here but using them anyway: a 4π rotation leave the 'macroscopic' electron body unchanged but changes 'microscopic' aspects of the electron? Here, the usual 'point' electron is the 'macroscopic' body and individual field lines are, or are related to, the 'microscopic' (sub-)parts of an electron?
[But .... when you do the waiter and tray version of the demonstration of 4π symmetry, your arm does not finish in a double twist, unlike your elastic strings ... how does that fit with your findings?]

This fits in with my mental image of the electron as having preon components. I don't see the field lines as exactly tracing out the paths of individual preons but there must be some relationship. E.g. an orthogonal relationship.

Field lines are supposed to join one -ve particle to a +ve particle. I recall Susskind in an online video saying that constitutes the conservation of charge. The field line must end up on a particle so we cannot remove one charged particle from the universe without contravening conservation. So the field lines connect particle A to particle B yet the wave contents of A start at particle A and end up at particle A at a later time.

Does a whole electron share all the field lines homogeneously? In my preons model the electron has different types of preons so could different field lines have different properties? In a simple analogy, one preon gives rise to one field line. And that one preon has its individual spin value, so I readily like the idea of a field lines having its own spin as 'microscopic' property of the electron.

In my preon model, all effects of a particle are caused by chiral properties. Even the properties of a scalar particle have a dynamic tension between opposite chiral parts. I have -ve electric charge being caused by chiral drive of preons into red, green and blue colour branes. and +ve charge being chiral drive into the anticolour branes. So a 'macroscopic' turn in spacetime of the whole point electron could weaken (or strengthen) the charge by weakening the overall effect of the 'microscopic spins' of the individual preons.

I do not see why this effect is only occurring at very low temperatures. At very low temperatures, are not all the special effects caused by bosonic pairs of electrons? (BECs require very low temperatures.)
Ben6993

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Re: Fractional Quantum Hall Effect

FrediFizzx wrote:Wow! I had never heard of this before. Perhaps this is something new.

Fred, what specifically? Not heard of FQHE? Or the twisting of entanglement threads even after orientation and entanglement are straightened out as in my experiment at https://jayryablon.files.wordpress.com/ ... -twist.pdf? I find it hard to think that nobody noticed this before, but I have been searching all over the web and not seen anybody talk about it at all. Jay
Yablon
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Re: Fractional Quantum Hall Effect

The twisting of threads. There may be something here in Schiller's Strand Model.

http://www.motionmountain.net/research.html

Been awhile since I read it but I don't immediately recall specifically what you show with the rubber band experiment above.
FrediFizzx
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Re: Fractional Quantum Hall Effect

Try this very short video.

No twists here at end of experiment?
Ben6993

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Re: Fractional Quantum Hall Effect

Ben6993 wrote:Try this very short video.
No twists here at end of experiment?

Fred wrote:The twisting of threads. There may be something here in Schiller's Strand Model.
http://www.motionmountain.net/research.html
Been awhile since I read it but I don't immediately recall specifically what you show with the rubber band experiment above.

Ben and Fred:

In http://commons.wikimedia.org/wiki/File: ... lement.gif they untangle using the thread on the right, twice. That gives two twists as I showed the other day. I did the experiment again, this time using the thread on the right for the first detangling step and then the thread on the left for the second detangling step. Now there are no twists at the end. So the method of how one untangles is "remembered" by nature (my pencils and rubber bands do exist in nature). I wonder if this is analogous to products of spin groups, i.e., SU(2)xSU(2) = spin 1 has a +1, 0 and -1 state, and maybe in a way previously unbeknownst, the mathematics can be used to record how you untangle things. +1 is two detangles with the left band, -1 is two with the right, and 0 is one each? Jay
Yablon
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Re: Fractional Quantum Hall Effect -- PREDICTION

I am comfortable enough to now claim that this paper at http://vixra.org/pdf/1411.0552v1.pdf contains a prediction, though until now I have not called it that:

The spin 1/2 state, I am saying, is a boson state with two electrons. So I predict that if somebody closely studies the 1/2 unit charge state, versus the 1/odd states:

a) if they find a way to discern the spin, they will find this the |Q|=1/2 state is a boson state with spin 1 and/or spin 0, and the 1/odd states are fermion states with spin 1/2.

b) if they find a way to test for compositeness of the |Q|=1/2 fractional state, they will see that this is a pair of quasi electrons, and that each carries a 1/4 unit of charge, but only when there is a close pairing of these quasi charges.

c) if they find a way to disrupt a |Q|=1/2 state, such as to move the electrons apart and break their pairing, it will degenerate into either two separate |Q|=1 electron states, or two |Q|=1/3 quasi particle states.

I would guess that there are experimental people who are clever enough to design methods to detect these results.

Jay
Yablon
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Re: Fractional Quantum Hall Effect

Hi Jay

Sorry for a very elementary question: are you referring to electric charge when you use Q in your latest post? Assuming that you are ...

The Q= 1/2 state could be a composite of two bits of two electrons but I would need to expand my preon model.
Temporarily and unfortunately ignoring weak isospin which does not get a mention, use (-0.25, -0.50) to refer to a new sub-unit of preons with electrical charge -0.25 and spin -0.5.
I already have an A block of preons with properties (-0.5, -0.5), so A1 would be a half of A but keeping all the spin but only half of the electric charge.

So for two bits of two LH electrons we get A1 + A1 = (-0.25, -0.50) + (-0.25, -0.50)
= (-0.5, -1) so we get Q = -0.5 and S = a bosonic value of -1.
I already had to sub-divide the C block into separate colour parts to explain the fractional electrical charges of the quarks. So having to sub-divide preon blocks A and B does not shock me as it seems to be a natural extension.

Assuming the two bits can only stay together as bits because of their close proximity and near zero temperature and 2D behaviour, when they are separated, you would get the two full LH electrons each with the property (-1, -0.5).

If you define B1 = (-0.25, +0.5), then
B1 + B1 = (-0.5, 1) giving bosonic spin +1. Ie adding two bits of two RH electrons.

Also, A1 + B1 = (-0.5, 0) giving bosonic spin 0. Ie take a bit of a LH electron and add it to a bit of a RH electron

Humourous footnote: every time I try to learn GR something stops me. Two weeks ago I started Susskind's online GR course. After one lecture I caught flu! On recovering from flu I find your new, fascinating paper on fractional charges which of course is a worthwhile distraction. The biggest distraction would take longer to explain: I bought Gravitation by M, T & W in 1971 and still have not read it properly.

All the best.
Ben6993

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Re: Fractional Quantum Hall Effect

Hi Jay

Some naive questions.

According to your paper, the windings with divisor 2 are for bosons which are paired electrons. (I amended my preon units to cope with that.) That is fine and fits in with Cooper pairs at low temperature although you take care not to require them to be Cooper pairs. Presumably that also includes cases such as winding = 4/2 giving Q=2? Presumably you require low temperature (even though temperature is not an explicit variable in your formulae?).

The other windings are for single electrons or part-electrons or quasi-fermions rather than bosons. So this is strange behaviour for non-bosons at low temperature. I cannot yet get anywhere near working out how to adapt my preon units to give fractional Q values which always have odd divisors for one preon unit, or a part of one unit. It is easy to get unrestricted fractional charges, though.

Do the windings correspond to rotations in S3 space (i.e. Joy's laboratory QED space, I trust) or are they windings in some abstract mathematical space. An electron undergoing a 4pi rotation in the S3 lab space will have all the properties it had before rotation? But a (quasi) electron undergoing an extra 4pi winding in whatever the space in your formulae, at low temperature, will undergo a change in fractional charge.

Best wishes.
Ben6993

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Re: Fractional Quantum Hall Effect -- PRD Review

Yablon wrote:In the process of doing this development, it occurred to me that certain solutions of the Dirac monopole had never been fully developed, and as I explored those states which had been "left on the table" by others, I began to realize that this could be used to account fully and completely for the fractionalized charge pattern observed in the Fractional Quantum Hall Effect. So over the weekend I wrote up an 11 page paper you may review at http://vixra.org/pdf/1411.0552v1.pdf, which as of Thursday I had no anticipation of writing. I do not think of myself as a condensed matter theorist, so I felt a bit like a fish out of water, but managed to do enough review of this area and got some private comments from a friend who is a professional condensed matter theorist, all of which made me comfortable I was not too badly off any bases. I also submitted this paper to PRD, and will let you know what happens.

I promised to share whatever happens, so let me do so. Erick Weinberg, the PRD Chief Editor, rejected this paper last week without review, saying that:
E. Weinberg / PRD wrote:This falls outside the scope of Physical Review D, and should instead be directed to a condensed matter journal.

I did not agree, and resubmitted with the following arguments:
Yablon wrote:I must respectfully record my disagreement with your dismissing my manuscript from consideration that DY11422 on the basis that it “falls outside the scope of Physical Review D, and should instead be directed to a condensed matter journal.”
At the link https://journals.aps.org/prd/about it is stated that “D15: covers general relativity, quantum theory of gravitation, cosmology, particle astrophysics, formal aspects of theory of particles and fields, general and formal development in gauge field theories and string theory.” This paper – if it is correct which you have not yet opined about – is most definitively an important “development in gauge field theories and string theory” which fits squarely within the scope of PRD.
The relationship which this paper bears to condensed matter rests in the fact that it demonstrates the power of gauge theory to explain a condensed matter phenomenon – namely the observed fractionalization of electric charge as observed under certain circumstances – strictly on the basis of formal gauge theory without resort to the condensed matter methodologies which are usually thought to be required for explanation.
Ever since Hermann Weyl developed gauge theory from 1918 to 1929 as an invariance under certain changes in phase with geometric roots in the curvature i.e. electromagnetic field of a gauge space, some of the most important theoretical advances in physics have been findings which extend the reach of gauge theories and demonstrate their power to explain a very broad range of observational data. Yang and Mills in 1954 extended the capacity of gauge theories by laying the foundation for these theories to eventually explain strong and weak interactions as factors in the phenomenological group SU(3)_C x S(2)_W x U(1)_Y. Weinberg and Salam, aided by developments in how to use the degrees of freedom in scalar fields to spontaneously break symmetry, explained the electroweak interaction on the basis of breaking S(2)_W x U(1)_Y down to the observed U(1)_em. Georgi and Glashow showed how SU(3)_C x S(2)_W x U(1)_Y could itself be obtained by breaking the symmetry of a larger simple group G which contains all of the strong, weak and electromagnetic interactions and the particles whose interactions they mediate, and used SU(5) as a demonstration of how this approach might work. Each of these developments were important because they showed how gauge theory could be extended to explain phenomenology well beyond what was known by Weyl and others back in 1929.
The specific focus of this paper, is to demonstrate that gauge theory is also powerful enough to be extended into the purview of phenomenon for which are thought to require a condensed matter explanation but which I show actually are grounded in gauge theory, and indeed, in the U(1)_em gauge theory of electromagnetism. The genealogical lineage of this approach starts with Dirac’s showing in 1931 that if one postulates a magnetic charge and uses a Dirac string connected from that charge to spatial infinity, then the electric charge must be quantized. (And I note that string theory is also within the scope of PRD.) In the mid-1970s Wu and Yang showed that the exact same result could be obtained without strings, using the indeterminacy at the north and south poles of an azimuth over a closed two-dimensional surface S^2 embedded in the three-dimensional space of physical experience. In Dirac’s original conception the gauge potential is defined everywhere except along the string over which it is indeterminate; Wu and Yang placed situated this indeterminacy at the north and south poles of S^2 and then absorbed this into a gauge transformation yielding the exact charge quantization that Dirac had found. What I demonstrate in this manuscript is that not only does this approach predict charge quantization, but it also predicts charge fractionalization. That is: gauge theory, when logically extended along the path laid out by Wu and Yang, predicts charge quantization and charge fractionalization.
It is certainly notable that the predicted fractionalization is observed in condensed matter environments of low temperature when magnetic fields are applied to certain materials. But what is notable is not that fractionalization is a condensed matter phenomenon, but that this condensed matter phenomenon is a logical consequence of gauge theory as applied by Wu and Yang, upon additional consideration of spinor orientation / entanglement relationships. Any advance which notably extends the reach of gauge theory beyond present understandings is within the scope of PRD. By demonstrating the power of gauge theory to be extended even further beyond its presently-known reach into a phenomenon presently thought to require a separate and distinct condensed matter explanation, this manuscript is most certainly within the scope of PRD.
Accordingly, I respectfully request reconsideration of your decision, and ask you to please review this work on its merits with view toward publication in PRD if it is found to be substantively sound.

I did not expect this to be accepted at PRD, but my goal in pressing was to force a substantive review, not just bureaucratic jousting over which journal covers what turf. Just this evening, I received another rejection, this time with the following review that is more substantive, and in my view, wrong:
E. Weinberg / PRD wrote:I find your arguments that this paper falls within the scope of Physical Review D unpersuasive. It is true that the scope of PRD is described as including gauge and string theories. However, the only gauge theory involved here is electromagnetism, and there was certainly no intention to claim that all papers involving electromagnetism should go to PRD rather than to PRA, PRB, or PRE. It is also true that the line of gauge singularities running from the monopole to infinity is often called a Dirac string, but that "string" has absolutely nothing to do with string theory.
In any case, the issue is really moot, because the paper is wrong:
1) If a monopole exists, then all particles carrying only electromagnetic charge must obey obey Eq. (1.7). They will therefore also trivially obey the weaker conditions of Eq. (2.2). However, the converse is certainly not true: obeying (2.2) but not (1.7) is not allowed. These statements have nothing to do with the spin of the particle.
2) Applying the Dirac-Wu-Yang arguments to the fractionally charged quasiparticles in FQHE systems is nontrivial. At the most fundamental level there is no issue. These systems are composed of electrons, protons, and neutrons, and there are no fractionally charged particles. At the level of analysis where the quasiparticle language applies, the system is fundamentally a two-dimensional one, whereas the Dirac-Wu-Yang argument is depends essentially on three-dimensional geometry.

The jurisdictional stuff is just chatter, let's get to the rejection points 1 and 2.

As to point 1, let me write down my equations (1.7) and (2.2):

$e=n\frac{2\pi }{\mu } =ne_{{\rm u}} =\frac{\Lambda }{\mu }$, (1.7)
$e=\frac{n}{m} \frac{2\pi }{\mu } =\frac{n}{m} e_{{\rm u}} =\nu e_{{\rm u}}$, (2.2)

where (2.2) goes together with the "filling factor''

$\nu \equiv \frac{n}{m} ;\quad n=0,\pm 1,\pm 2,\pm 3...;\quad m=1,2,3,4,5,6...$. (2.3)

I agree with E. Weinberg that "If a monopole exists, then all particles carrying only electromagnetic charge must obey obey Eq. (1.7)." I also agree that "They will therefore also obey the conditions of Eq. (2.2)," but the words "trivially" and "weaker" are gratuitous. It is important to understand that the DCQ predicts charge fractionalization. As far as I know nobody has noticed this before, and that is not "trivial." Perhaps the (2.2) conditions are "weaker" in that they permit more charge states than does the quantized $ne_{{\rm u}}$, but if those fractionalized charge states happen to be observed anywhere in the physical world under any set of circumstance however limited, then whether you call them weaker or call them Frisbees, they are important to understand. I also agree with E. Weinberg that "the converse is certainly not true: obeying (2.2) but not (1.7) is not allowed." I have not said it is allowed. The m=1 state in my (2.2), see (2.3), is precisely (1.7). (2.2) includes (1.7) as its m=1 subset. If I am missing something that Weinberg is trying to say, please help me out. The final statement that "These statements have nothing to do with the spin of the particle" goes to the heart of how we interpret these results, and that leads to point 2.

Point 2 goes to the heart of the matter, and it was this sort of review I wanted to extract from E. Weinberg because it helps me understand what I need to explain to potential readers. I totally agree that "Applying the Dirac-Wu-Yang arguments to the fractionally charged quasiparticles in FQHE systems is nontrivial." I read his statement that "At the most fundamental level there is no issue" to really be saying that at the most fundamental level there is no connection. If that is so, then that is precisely what I am trying to contradict with this paper. I believe there is a connection. I agree that when Dirac uses strings to predict the DQC or the Dirac monopoles, or when Wu and Yang do the same thing without strings, they are working with "systems ... composed of electrons, protons, and neutrons, and there are no fractionally charged particles."

But we also know that under certain circumstances, at ultra-low temperatures, and inside of certain materials, as Weinberg correctly states, "the system is fundamentally a two-dimensional one, whereas the Dirac-Wu-Yang argument is depends essentially on three-dimensional geometry." This is, indeed, "at the level of analysis where the quasiparticle language applies." Weinberg's conclusion is that because of this, there is no connection, i.e., no "issue." My contrary conclusion is that there is a real connection, and that connection needs to be understood.

My paper spells out the role of orientation / entanglement in this connection, so I will not repeat that here in any detail. But think about what is physically happening when you take a Dirac-Wu-Yang gauge theory problem in three space dimensions, turn the temperature way down so as to "clamp down" on the electrons, and force them into a two dimensional system where the material employed has removed a spatial degree of freedom. In that that two dimensional, ultra-cold system, the environment of the electrons is bearing down on them in a way that does not ordinarily occur in warmer three-dimensional systems. Misner, Throne and Wheeler teach that electrons are entangled with their environment, even in three dimensions. But there is clearly something about the low temperature and the two-dimensional restriction which causes the entanglement to manifest itself in what is physically observed. One might envision that in three dimensions and at higher temperatures, whatever "entanglement" an electron might find itself in is easily straightened out and so not readily observed in physical, observable manifestation. But when you cool down enough and take away a degree of freedom, you have "cornered" or "trapped" the electron and now it finds itself in states where its entanglement relationships are physically observed via FQHE with the filling factors

$\nu =\frac{\rlap{--}\Lambda }{\rlap{--}\phi } ;\quad \rlap{--}\Lambda =0,\pm 1,\pm 2,\pm 3...;$
$\rlap{--}\phi =2$ --or-- $\rlap{--}\phi =1+2m;\quad m=0,1,2,3...$ (3.2).

That also means, as I state at the end of my section 3, that "to the degree that the filling factors (3.2) do describe a feature of the natural world but only under these specialized conditions, and because (3.3) [for $\oint F =\nu \mu _{{\rm u}}$] is integrally related to (3.2), it would appear that the non-zero magnetic fluxes $\oint F =\nu \mu _{{\rm u}}$ of Dirac monopoles (as distinguished from other types of monopole) would only evidence themselves in nature under equally-restricted conditions."

Let me put this a bit differently. The Dirac-Wu-Yang arguments do predict that if magnetic monopoles exist, then both electric and magnetic charges are quantized, and as I have shown, they also predict charge fractionalization, and if we only count the states that differ by a $4\pi$ azimuth, these will have odd-denominators-only. Now, in the three-dimensional space of Dirac-Wu-Yang arguments none of these is observed with the exception of charge quantization for electric charges only. There are no magnetic monopoles, there is no fractionalization, and the modern-day understanding of charge quantization is not based on the DQC, but is based on the Yang-Mills generator Q=Y/2+I^3 which arises from spontaneous symmetry breaking. So: Dirac-Wu-Yang is elegant and beautiful but wrong (i.e., not physically observed), except at "the level of analysis where the quasiparticle language applies [and] the system is fundamentally a two-dimensional one." There, and only there, nature puts on display the logical consequences of Dirac-Wu-Yang, and Dirac-Wu-Yang are now shown to be correct, in the special environment that gives rise to the FQHE. For, in this very special environment where electrons are chilled way down in a material which confines them to two dimensions, all of a sudden the fill factors (3.2) above emerging from Dirac-Wu-Yang with entanglement and with boson pairing of unentangled charge states are exactly what is observed. Entanglement tells us that the electrons are intimately connected with their environment and it should not be too much of a conceptual leap to connect this to the "collective" "many-body" approach of condensed matter physics. As a matter of scientific openness and discourse, these results should certainly be made known so that there is an opportunity for these connections to be understood and tightened.

So, in the end, gauge theory taken together with entanglement is shown to have the power to explain a phenomenon in condensed matter land which E. Weinberg and others do not yet understand it has the power to do.

I live to fight another day!

I do want to say one other thing, now that I am disclosing openly that Erick Weinberg is the person with whom I have been "arguing" about my work for the past year and a half, trying to be allowed in past the "gate" that he and his colleagues maintain about the physics fortress. He is the person with whom I spent 90 minutes talking at the APS meeting in April. And I have so far not convinced him of anything, but he has convinced me that he will be unremittingly critical and this is of inestimable value to me. That is why I keep going back for what I expect will be more harsh critique. I respect that, and I know that that is the only way to ultimately learn how to present my work in a way that it will start to be recognized. Baryons are the magnetic monopoles of Yang-Mills gauge theory and one day that will be known by every student who takes nuclear physics 101; but Weinberg showed me that it is the t'Hooft-Polyakov version of the Yang-Mills monopole and the scalar fields which provide stability to these monopoles that I have been talking about all along, which I need to make the pedagogically clear. And he has now shown me that I have to lay out a sensible path for people wearing field theory hats and condensed matter hats to understand my FQHE results and how they cross between these two areas for which they believe "at the most fundamental level there is no issue." With all due respect, at the most fundamental level, there is a deep connection.

Jay
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Re: Fractional Quantum Hall Effect

Jay, the great majority of your material is couched in language and maths beyond my scope to handle. However, I'd like clarification on one basic point please. A Dirac-type monopole is essentially just one end of a notionally infinitely long filimentary magnetic flux string. As such, the string is the real object and the 'monopole' is an artifice which only 'exists' via discounting the string return flux when performing a Gauss surface integral of net flux enclosing the 'monoople'.

The genuine dual type of monopole, as traditionally proposed by the extended Maxwell's eqns - e.g. https://en.wikipedia.org/wiki/Magnetic_ ... _equations
is a genuine point particle source of net Coulombic magnetic flux. And I have made basic comment on what such a genuine monopole would imply here:
viewtopic.php?f=3&t=65
Now I understand that your monopoles are confined and that suggests 'massive photons' or some-such as force carriers. Nevertheless, their basic character must I would think be of either the Dirac type or the genuine dual type (with net enclosed flux different but non-constant functions of radius in either of such 'confined' cases). Which is it?
Q-reeus

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Re: Fractional Quantum Hall Effect

Jay

Your critic said that the Dirac Strings were not the strings of string theory. My first reaction to you referring to them wrt string theory (quite a while ago, somewhere) was also to think "but they are not the same as 'string theory' strings". However, I am unsure of that and also unsure of the definition of a string. I followed an online course by Susskind so I have some idea, but you will no doubt know much more than me about them. I suppose strings are entities which fit the mathematical descriptions of them in string theory. Which is not very helpful for visualisation, but maybe rules out Dirac strings. At one point Susskind, if I remember correctly, described strings early as having quarks at their ends. And if you break the string, you get new quarks at the new ends. By the finish of the course though, I was visualising a string as a long entity which had either one or both ends on a colour brane, even for QED particles despite colour seeming to be a QCD property, and could be an open string or a closed string. They also require speed c.

I like your idea of field lines (Dirac strings) having spin properties of their own which implies they are not intangibles. I visualise field lines as tangibles closely related to the contents of particles in wave or field form. And since an electron wave function is present, AFAIK, everywhere i.e. not just in one stringlike thread, then the electron's dispersed contents can give rise to both electric and orthogonal magnetic field lines simultaneously. And since my description has preons starting on a colour brane and moving at speed c, and having chirality, then my preons are 'string theory' strings and can also fit as Dirac strings? Implying that Dirac strings could be 'string theory' strings?

I noticed a 2010 arxiv paper on Dirac strings (or, rather, on real classical analogues of the unbservable Dirac strings) at http://arxiv.org/abs/1011.1174. On pages 8 and 9 it mentions 3D strings rather than 2D. I think you are saying that the fractional effect is not just a condensed matter 2D effect, so this paper may be relevant. On page 8, and earlier, it mentions reverse spins in tensionless Dirac strings. At some point you mention that field lines at cold temperatures may become loosened from the monopoles. Maybe the looseness/ tensionlessness(?) allows the reverse spin?

The term 'reverse spin' reminds me of spin bowling in cricket, which gives a sporting analogy. Assume slow bowling relates to low temperatures and fast bowling relates to high temperatures. Slow bowlers can obtain (reverse) spin while fast bowlers do not get spin in cricket.

Please excuse me for rambling on.
Ben6993

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Re: Fractional Quantum Hall Effect

Q-reeus wrote:Jay, the great majority of your material is couched in language and maths beyond my scope to handle. However, I'd like clarification on one basic point please. A Dirac-type monopole is essentially just one end of a notionally infinitely long filimentary magnetic flux string. As such, the string is the real object and the 'monopole' is an artifice which only 'exists' via discounting the string return flux when performing a Gauss surface integral of net flux enclosing the 'monoople'.

The genuine dual type of monopole, as traditionally proposed by the extended Maxwell's eqns - e.g. https://en.wikipedia.org/wiki/Magnetic_ ... _equations
is a genuine point particle source of net Coulombic magnetic flux. And I have made basic comment on what such a genuine monopole would imply here:
viewtopic.php?f=3&t=65
Now I understand that your monopoles are confined and that suggests 'massive photons' or some-such as force carriers. Nevertheless, their basic character must I would think be of either the Dirac type or the genuine dual type (with net enclosed flux different but non-constant functions of radius in either of such 'confined' cases). Which is it?

Hi Q-reeus:

I have actually been working to clarify that same basic point in my own mind, so that I can then clarify it for everybody else.

The monopoles I have been working on since the Spring of 2005 when I first formed the belief that baryons are Yang-Mills monopoles, are what you call the "genuine dual type of monopole," but they only come about because of the fact that gauge fields do not commute in Yang-Mills gauge theory, $\left[G_{\mu } ,G_{\nu } \right]\ne 0$. What I have come to understand in the almost ten years I have worked on this is that these monopoles do not, however, become topologically stable without emerging via spontaneous symmetry breaking using Higgs scalar field, and although I have pointed this out in my papers, what I learned from Weinberg's review was that my pedagogical presentation of this is not yet in the form that it will register properly on the people who hand out admission tickets to the top echelons of the physics world. And the reason for this is that I need to place the Higgs field front and center and show its exact use and operation in my theory to provide this topological stability.

To do this, in my next paper I will work with a roadmap from pages 469 to 475 of Cheng and Li, which I have uploaded to https://jayryablon.files.wordpress.com/ ... 69-475.pdf. (Unfortunately the page numbers were cut off in the scan.) I will show that the t'Hooft monopoles, generalized to the GUT "colour magnetic charges" arrived at from Georgi-Glasgow SU(5) and referenced on 473 of this link, are sufficient to support my monopole theory in its entirety. There is more, of course, but I need to fly close to the standard model to get these guys to understand this, so that is what I will do.

The key connection is made in (15.87) of the above linked file, which is important enough to reproduce here:

$\begin{array}{l} {F_{3} ^{ij} =\partial ^{i} A^{j} _{3} -\partial ^{j} A^{i} _{3} -e\left(A^{i} _{1} A^{j} _{2} -A_{1} ^{j} A_{2} ^{i} \right)} \\ {\quad \; \; =\partial ^{i} A^{j} _{3} -\partial ^{j} A^{i} _{3} +\frac{1}{ea^{3} } {\bf \varphi }\cdot \left(\partial ^{i} {\bf \varphi }\times \partial ^{j} {\bf \varphi }\right)} \end{array}$ (Cheng and Li 15.78)

It will be recognized that the term $\left(A^{i} _{1} A^{j} _{2} -A_{1} ^{j} A_{2} ^{i} \right)}$ is just one special case of $\left[G_{\mu } ,G_{\nu } \right]\ne 0$ which I have always used as the heart of my monopoles $\oint F =-i\oint \left[G,G\right] \ne 0$. But my pedagogical sin is that I have not been explicitly paying homage to the term ${\bf \varphi }\cdot \left(\partial ^{i} {\bf \varphi }\times \partial ^{j} {\bf \varphi }\right)}$ which comes from the symmetry breaking using Higgs scalars that leads to the t'Hooft monopole. All of Weinberg's critique referring to the (absence, in his view, of) scalars, was because I was not showing the connections in (15.78) above. So from here I will.

So, as a warmup for my paper taking another run at my monopole thesis using the pedagogy these guys want to see, I went back to the Dirac monopoles, which you correctly point out, are not "genuine" in a certain respect, just to use them in the introduction of my paper. I had nothing special planned for the Dirac monopoles other than to set the context for the rest of the paper.

Now, I have looked at the Wu and Yang derivation of Dirac monopoles in https://jayryablon.files.wordpress.com/ ... opoles.pdf hundreds of times and know it in my sleep, but when I went to write it up, it struck me that the derivation was leaving out certain rotational states, and that when they were included, the charge became fractional. So of course I made the leap to ask whether this could connect with FQHE, but needed odd denominator fractions, and saw that orientation-entanglement was the way to get those. So for now, I will spend a week or two focusing on this paper to get it right before I return to my Yang-Mills (now t'Hooft-Polyakov) monoples, because I have already laid out the key discoveries there and am just writing to get the big shots to see those, whereas what I am now discovering about condensed matter physics is new to me and will be new to the world and is not yet written up. I always give priority to getting new stuff down on paper, over cleaning up the stuff I have already written even if the latter is needed to satisfy the grand pooh-bahs.

So to get back to your question: the Yang Mills / t'Hoof-Polyakov monopoles are the real deal "genuine" monopoles which are protons and neutrons and baryons, including as you correctly deduce, fluxes which are "non-constant functions of radius" and indeed are the mainspring of the short range of the nuclear force. The Dirac monopoles are not observed, except when, as I have now discovered in the past two weeks, you get the electrons cooled down to near zero degrees K and force them into a two dimensional configuration so that they cannot untangle as they otherwise would and thus show fractional charges and enter a state of electric-magnetic duality symmetry for Maxwell's U(1)_em equations. Put another way, just as certain symmetries exist at GUT energies and are broken at lower energies, there are also certain symmetries that exist near zero degrees K in materials that constrain electrons, and the granddaddy of those symmetries is the EM duality which quickly gets broken if you raise the temperature or give the electrons too much 3-D freedom. But in that very limited environment, there is an electric magnetic duality for Maxwell's equations themselves, and moreover, as I have found in just these past 48 hours and plan to write up this weekend, the azimuth quantum numbers which tell you the fractional denominator in FQHE, I believe, are directly related to the azimuth quantum numbers for electronic states in atomic shells, i.e., the orbital number l and the magnetic number $-l\le m\le l$.

This will lead me to propose an experiment to the condensed matter guys: Generate so-called quasi-particles with |Q|=1/3. Study those quasi-particles the way you would study an electron in an atomic shell. I predict that aside from their reduced charges, they will show characteristics of l=1 electrons with m=-1,0,+1. Then create the |Q|=1/5 quasi particles. Or, because you need really strong magnetic fields to do 1/5, if you can't do that, settle for 2/5. I predict that aside from their reduced charges, these will show characteristics of l=2 electrons with m=-2,-1,0,+1,+2. If I am right, that will confirm Dirac monopoles in condensed matter at low energies. I get to this through orientation/entanglement in a way I will carefully lay out, through the topological twisting of the entanglement threads (which as I have said has previously been either overlooked or ignored in the literature to the best of my knowledge), which is illustrated at https://jayryablon.files.wordpress.com/ ... figure.jpg which I will spruce up into a Figure in this paper. (I did not go to NSF for the funding to put together this experiment; I sprung for the five bucks of supplies myself ) These photos are a clue: look at them closely, and use them to figure out how I get to this prediction. (Hint: look at the number of electrons that get into s,p,d,f... shells in an atom, http://en.wikipedia.org/wiki/Electron_shell, look at the number of twist configurations associated with a given orientation of the wood pieces and compare, and notice how my azimuth is defined in the same physical space and in the same way as the azimuth of the orbital quantum number. This turns into a study of the topology underlying the atomic quantum numbers, and it is condensed matter that reveals this to us.)

I never thought of myself as a nuclear theorist until two years ago when I found binding energies and needed to explain them with a nuclear theory. I never thought of myself as a condensed matter theorist until two weeks ago when I found FQHE and now need to explain all of this with a theory of atomic shells in condensed matter.

Jay
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Re: Fractional Quantum Hall Effect

Yablon wrote:Hi Q-reeus:
I have actually been working to clarify that same basic point in my own mind, so that I can then clarify it for everybody else.

Some nice timing then.
The monopoles I have been working on since the Spring of 2005 when I first formed the belief that baryons are Yang-Mills monopoles, are what you call the "genuine dual type of monopole," but they only come about because of the fact that gauge fields do not commute in Yang-Mills gauge theory, $\left[G_{\mu } ,G_{\nu } \right]\ne 0$.

I'm not up with the commutative/anti-commutative mathematical properties, but good to get that clarification re genuine monopoles.
What I have come to understand in the almost ten years I have worked on this is that these monopoles do not, however, become topologically stable without emerging via spontaneous symmetry breaking using Higgs scalar field, and although I have pointed this out in my papers, what I learned from Weinberg's review was that my pedagogical presentation of this is not yet in the form that it will register properly on the people who hand out admission tickets to the top echelons of the physics world. And the reason for this is that I need to place the Higgs field front and center and show its exact use and operation in my theory to provide this topological stability.

Higgs seems to have firmly taken center stage in particle physics, despite recent claims what LHC found may well be something else:
http://www.sciencetimes.com/articles/11 ... ticles.htm Majority favors Higgs but only time will tell.
To do this, in my next paper I will work with a roadmap from pages 469 to 475 of Cheng and Li, which I have uploaded to https://jayryablon.files.wordpress.com/ ... 69-475.pdf. (Unfortunately the page numbers were cut off in the scan.) I will show that the t'Hooft monopoles, generalized to the GUT "colour magnetic charges" arrived at from Georgi-Glasgow SU(5) and referenced on 473 of this link, are sufficient to support my monopole theory in its entirety. There is more, of course, but I need to fly close to the standard model to get these guys to understand this, so that is what I will do.

Ah, so i gather your monopoles are generalizations and/or analogues of the original purely EM variety. Sort of suspected so but wasn't up to discerning that from your higher level material.
...So, as a warmup for my paper taking another run at my monopole thesis using the pedagogy these guys want to see, I went back to the Dirac monopoles, which you correctly point out, are not "genuine" in a certain respect, just to use them in the introduction of my paper. I had nothing special planned for the Dirac monopoles other than to set the context for the rest of the paper.

Well, I basically got from a long-ago read of Dirac's original paper, that somehow the flux-tube/string was to be considered an unobservable. Strange in a number of ways, but somehow Dirac got away with using it. For one thing, the energy-momentum locked up in the string should by elementary argument be vastly greater than in the relatively minor 'end leakage' aka 'monopole'. Implying the dynamics of a Dirac monopole should be that of a massive flux tube and not a point particle of any assumed mass. And then you have to ask about the implied stresses present in the flux-tube/string and how there can be stability of such - i.e. would it spontaneously curl up or writhe or require relativistic motions or what? Ignoring all such 'minor issues' such model nevertheless is evidently still ok for certain situations - say dynamics of an interacting electron that never intersects the string. But that model would founder if applied to the monopole-supercurrent scenario in that thread earlier linked to. As to whether the source q_m is true (derives from a magnetic scalar potential) or Dirac-type (magnetic vector potential) is then crucial. Such thoughts though probably have zero bearing on your newly discovered low K quasi-particle regime.
Now, I have looked at the Wu and Yang derivation of Dirac monopoles in https://jayryablon.files.wordpress.com/ ... oles.pdf...

Regarding the comment made in 2nd para following eqn (10) there:
"Note that the whole point is that F is locally but not globally exact; otherwise by (6) the magnetic charge g = int_S^2 F would be zero."
My translation of that: "i.e. ignore the string internal flux. In fact ignore everything about the string except one of it's ends!"
...I always give priority to getting new stuff down on paper, over cleaning up the stuff I have already written even if the latter is needed to satisfy the grand pooh-bahs.

Yes, the time-proven maxim: 'write it down at the time, lest it gets forgotten!'
So to get back to your question: the Yang Mills / t'Hoof-Polyakov monopoles are the real deal "genuine" monopoles which are protons and neutrons and baryons, including as you correctly deduce, fluxes which are "non-constant functions of radius" and indeed are the mainspring of the short range of the nuclear force. The Dirac monopoles are not observed, except when, as I have now discovered in the past two weeks, you get the electrons cooled down to near zero degrees K and force them into a two dimensional configuration so that they cannot untangle as they otherwise would and thus show fractional charges and enter a state of electric-magnetic duality symmetry for Maxwell's U(1)_em equations. Put another way, just as certain symmetries exist at GUT energies and are broken at lower energies, there are also certain symmetries that exist near zero degrees K in materials that constrain electrons, and the granddaddy of those symmetries is the EM duality which quickly gets broken if you raise the temperature or give the electrons too much 3-D freedom. But in that very limited environment, there is an electric magnetic duality for Maxwell's equations themselves, and moreover, as I have found in just these past 48 hours and plan to write up this weekend, the azimuth quantum numbers which tell you the fractional denominator in FQHE, I believe, are directly related to the azimuth quantum numbers for electronic states in atomic shells, i.e., the orbital number l and the magnetic number $-l\le m\le l$.

I take it such Dirac monopoles are linked by their attendant flux strings. Would such strings lie in the 2D planes containing the monopoles, or connect mainly via the 3rd D?
This will lead me to propose an experiment to the condensed matter guys: Generate so-called quasi-particles with |Q|=1/3. Study those quasi-particles the way you would study an electron in an atomic shell. I predict that aside from their reduced charges, they will show characteristics of l=1 electrons with m=-1,0,+1. Then create the |Q|=1/5 quasi particles. Or, because you need really strong magnetic fields to do 1/5, if you can't do that, settle for 2/5. I predict that aside from their reduced charges, these will show characteristics of l=2 electrons with m=-2,-1,0,+1,+2. If I am right, that will confirm Dirac monopoles in condensed matter at low energies. I get to this through orientation/entanglement in a way I will carefully lay out, through the topological twisting of the entanglement threads (which as I have said has previously been either overlooked or ignored in the literature to the best of my knowledge), which is illustrated at https://jayryablon.files.wordpress.com/ ... figure.jpg which I will spruce up into a Figure in this paper. (I did not go to NSF for the funding to put together this experiment; I sprung for the five bucks of supplies myself )

And current version is a nice test for color blindness to boot! Seriously, for an extra buck Jay, consider marking all the way along one edge of each ribbon in say black. Much easier to then figure out the sense and degree of twist. But I guess you are aware of that and it will all look different in 'final edition'.
...These photos are a clue: look at them closely, and use them to figure out how I get to this prediction. (Hint: look at the number of electrons that get into s,p,d,f... shells in an atom, http://en.wikipedia.org/wiki/Electron_shell, look at the number of twist configurations associated with a given orientation of the wood pieces and compare, and notice how my azimuth is defined in the same physical space and in the same way as the azimuth of the orbital quantum number. This turns into a study of the topology underlying the atomic quantum numbers, and it is condensed matter that reveals this to us.)

Lots of avenues to follow there. What I can somewhat follow reminds of someone into braids I came across re the LQG crowd here: https://en.wikipedia.org/wiki/Sundance_Bilson-Thompson
Seems to have a lot in common with your own thinking. Maybe you have or might want to communicate with such folks.
I never thought of myself as a nuclear theorist until two years ago when I found binding energies and needed to explain them with a nuclear theory. I never thought of myself as a condensed matter theorist until two weeks ago when I found FQHE and now need to explain all of this with a theory of atomic shells in condensed matter.

Exciting and doubtless difficult times ahead Jay. Hoping your renewed efforts achieve that final breakthrough against entrenched views.
Q-reeus

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Re: Fractional Quantum Hall Effect

Q-reeus wrote:And current version is a nice test for color blindness to boot! Seriously, for an extra buck Jay, consider marking all the way along one edge of each ribbon in say black. Much easier to then figure out the sense and degree of twist. But I guess you are aware of that and it will all look different in 'final edition'.

Limited time today, so I'll reply for now on just this on point. I spent twenty minutes on Lowe's Hardware with 3 different sales people trying to help me find what I described as a cross between a shoelace and a ribbon for a "science demonstration" and not any home improvement. The best we all could get was an orange ribbon that is used to cordon off areas at a construction site. So I brought the ribbon home intending to mark one side black and leave the other side orange. But the ribbon was so thin that the black marker bled through and it looked black on both sides. So I threw that away and used a white wall primer to paint one side white while leaving the other side orange. And that is what you saw in the picture I posted. And, by the way, I myself do happen to be color blind.
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Re: Fractional Quantum Hall Effect

Ben6993 wrote:Jay
Your critic said that the Dirac Strings were not the strings of string theory.

Ben,

I'll stop you right there. On that point, my critic was correct. I threw in the mention of strings parenthetically, because I was miffed that they would not review the paper and just proclaimed that it was "outside the scope" of PRD which is clearly baloney. It may be within the scope of other journals, but that is not the same as being outside the PRD scope. So I fought their baloney with my own baloney, playing with the word "strings," which in retrospect was bad judgment because it is always best to keep the high ground. I did read somewhere that strings came out of Dirac strings, but personally, I don't really see that and agree with my critic on that point. I did get what I wanted in the end, which was Weinberg's substantive comments. So all the jurisdictional stuff, as he also says, is moot.

Jay

PS: I do think that strings are very relevant in a preon model, especially two preons which I have used in the past, because the ends of the strings provide a means to connect the preons.
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