Completed Fractional Quantum Hall Paper

[Yablon]

Dear Friends:

As you know I have been working steadily during the last five weeks on the Fractional Quantum Hall Effect (FQHE). My paper on this is now complete and just last evening I submitted it to a well-known and well-regarded journal. I will not say which one at this time, but I will say that it is not part of the APS system. You may read this at:

http://vixra.org/pdf/1412.0267v1.pdf

I earlier posted the E. Weinberg critique of my initial paper on this subject. The present paper fully and thoroughly answers that critique, and goes well beyond. The two main points that Weinberg made at the time regarded a) the indistinguishability of orientations differing by 2pi (he said that equating all solutions differing by a 2pi orientation was "trivial"), and b) the two-dimensionality of the FQHE system in contrast to the presumed three-dimensionality of the Dirac-Wu-Yang (QWY) derivation. These were not fatal problems because I have fully addressed them here, but these were legitimate critiques because they pointed out to me the questions I would be required by others to answer in order to have this work favorably recognized. So I am glad that I pressed him to provide that review.

The one critique that he did not make, which I have since made of my own work, and which is even more important that the two that he did make, is that I did not originally show how you could have magnetic monopoles near 0K leading to FQHE, and yet not have magnetic monopoles at temperatures much higher than 0K. In short, I had not really solved the "monopole problem" that is endemic to the Dirac Quantization Condition (DQC) and that has been on the table really since Maxwell's day, by showing how to break the low temperature electric-magnetic duality so that you could have monopoles at ultra-low temperatures, but have them gone at higher temperatures.

I only really solved this duality symmetry breaking problem in the past several days, and Section 9 of this paper presents this solution publicly for the first time. If you have followed this work as I was progressing, then section 9 here is what is completely new in relation to anything you have seen before. What I have found is that at higher temperatures moving up from 0K, the magnetic monopoles do become zero, but they are replaced by a "thermal residue" which appears to be responsible at the microscopic level the the very existence of heat in the universe. In this way, Section 9 may be the start of a unification of electromagnetic gauge theory with thermodynamics.

As you know I have maintained extensively and continue to maintain the the magnetic monopoles of Yang-Mills gauge theory, in t'Hooft-Polyakov form following symmetry breaking, are the observed protons and neutrons including baryons. But these monopoles are not the U(1) monopoles of Maxwell. They are SU(3) colored monopoles of non-Abelian gauge theories.

The monopole problem I am talking about presently in FQHE and DQC is the original magnetic monopole problem dating back to Maxwell. The magnetic monopoles I am dealing with here are the true, original magnetic monopoles that Maxwell's theory does not contain because it lacks duality symmetry. What this paper demonstrates is that in the real physical world we inhabit, the absence of monopoles means the presence of heat, and the presence of monopoles means the absence of heat.

Jay

[Joy Christian]

Congratulations, Jay.

The paper now looks like a solid piece of work. Good luck with the publication process.

Best,

Joy

[Yablon]

Congratulations, Jay.

The paper now looks like a solid piece of work. Good luck with the publication process.

Best,

Joy

Thanks Joy, hoping for the best. I am very pleased to have been able to bring orientation / entanglement (OE) topology to a new level of mapping atomic structure and heat.

Finding and fixing the hidden assumption in the Wu-Yang derivation of Dirac monopoles as I finally managed to do in section 9 was not easy, because I have looked at and thought about that derivation hundreds of times over the past five years to the point that I knew it in my sleep, and it is always very hard to see something differently when it is that familiar to you.

I would not have been able to ferret out the answer without carefully studying OE and using that to find out what I had to be looking for. Topology is a wonderful guiding light in dark corners of physics. Once I realized that 0K had to be characterized by the total freezing of the OE "object" and also its "threads" relative to the "environment" and that this is topologically why there IS an absolute temperature minimum in the first place, I knew that I had to go back to the gauge theory and look for some place where we are assuming that some things are frozen in lock step with one another, when the assumption of this freezing is only exactly correct at 0K. Then it hit me that when we treat the north gauge field patch as differing from the south gauge field patch by no more than a gauge transformation, we are "freezing" them together which looks nice geometrically, but physically is only valid at 0K. But if the north and south patches differ by more then a gauge transformation -- which means the difference has to be an observable -- and because the gauge field is an energy in the form of an electromagnetic potential, then whatever it is that defines their difference also has to be an energy, but an energy in the form of something other than an electromagnetic potential. Because the form of energy which brought me to this dance via the guiding light of topology was heat (really, taking away all the heat), I knew that this energy which broke off the ability to gauge transform between the two patches had to represent heat. (And it just occurred to me this very moment that this is a new way to break a gauge symmetry.) And that broke me through the wall of my own and everybody else's mental constructs to find the hidden assumption in the Dirac-Wu-Yang derivation that nobody realized was a hidden assumption. After all, if this derivation was predicting magnetic monopoles and we are not observing magnetic monopoles, then however pretty the derivation is, there is something wrong with it and we have to find out what that is. And what that is, is that this derivation is only physically valid, exactly, at 0K. A bit higher it is approximate and a bit higher than that the EM duality breaks, and the monopoles disappear and are replaced by a heat function for which the energy that separates the gauge patches is the spacetime gradient.

The net result is what I believe will turn into a unification of electromagnetic gauge theory and thermodynamics at the microscopic level, running right through magnetic monopoles. And along the way, we also bag a topological model of atomic and nuclear shell quantization.

Jay

[Ben6993]

Hi Jay

I have read your section 9 (reading it as best I can and much will have gone over my head). The thermodynamics connection is wonderful ... I wasn't expecting that!

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The paper seems excellent and too deep for me to comment on, but a few questions anyway

At 0K, would the magnetic monopoles have fractional magnetic charges, too?

Calling the effect heat seemed odd to me at first but I suppose it is fine (I haven't read any thermodynamics since first year at university (1967/8) so maybe I am biased against it!). All the heat produced in this way, for large temperature, is tied up [at least initially] inside the electrons' energy levels? My first thought was it was an "energy level" contribution to the total mass, as I suppose it cannot be rest mass, and not simply KE. So the higgs spontaneous symmetry breaking gives rest mass to the electron, while your [spontaneous symmetry breaking?] mechanism adds energy levels for the electrons?

Your formulae, if I read them correctly, apply also to single electrons at 0K. At 0K there is maybe not enough energy to put an electron in an outer quantum number. So having a fractional charge allows packing the first shell with many electrons all with different fractional charges whereas at high temperature there could only be two electrons in that first shell? Or do I have it wrong?

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All the best with getting the paper published.

[Yablon]

Hi Jay

I have read your section 9 (reading it as best I can and much will have gone over my head). The thermodynamics connection is wonderful ... I wasn't expecting that!

Ben

Neither was I. Its been a few years since I took a good look at thermodynamics, but I am dusting it off again to try to connect all of this to internal energy (accumulated from heat and work) and just this afternoon derived what is sort of a Maxwell's equations version of thermodynamics that I may be able to connect to all of this. To find entropy, I need something that always increases over time, and I think I have found it. But stay tuned...

="Ben6993"
The paper seems excellent and too deep for me to comment on, but a few questions anyway

At 0K, would the magnetic monopoles have fractional magnetic charges, too?

Yes, that is part of what I am saying.

Calling the effect heat seemed odd to me at first but I suppose it is fine (I haven't read any thermodynamics since first year at university (1967/8) so maybe I am biased against it!). All the heat produced in this way, for large temperature, is tied up [at least initially] inside the electrons' energy levels? My first thought was it was an "energy level" contribution to the total mass, as I suppose it cannot be rest mass, and not simply KE. So the higgs spontaneous symmetry breaking gives rest mass to the electron, while your [spontaneous symmetry breaking?] mechanism adds energy levels for the electrons?

More or less. I am still not sure how this all plays out, because the temperature is usually described in terms of a statistical mean of kinetic energy from vibratory motions of particles, but here, I am treating the internal energy as intrinsic to the electron albeit zero at 0K and higher at larger temperature. I am going to need some time to reflect on my unanticipated entry into thermodynamics just as I needed time six weeks ago to sort out my unexpected entry into condensed matter when all I was trying to do was mount a defense of my baryons = YM monopoles theory. Those things happen if you are following nature rather trying to have nature follow you.

Your formulae, if I read them correctly, apply also to single electrons at 0K.

Yes.

At 0K there is maybe not enough energy to put an electron in an outer quantum number. So having a fractional charge allows packing the first shell with many electrons all with different fractional charges whereas at high temperature there could only be two electrons in that first shell? Or do I have it wrong?

Wrong. Even at 0K Exclusion still applies. The higher-integer denominator states are in elevated energy states. I assume that these will map over to things like Fermi energies or Landau Levels, but condensed matter is till new to me so there too it may take some time to make all of the connections directly that I do believe will be found here.

All the best with getting the paper published.

Thanks!

And happy new year to everybody!

Jay

[Ben6993]

Jay wrote:
... the temperature is usually described in terms of a statistical mean of kinetic energy from vibratory motions of particles, but here, I am treating the internal energy as intrinsic to the electron albeit zero at 0K and higher at larger temperature. ...

In terms of dice-rolling games, you have defined all the infinite possible values to go on the faces of the dice[=electrons]. But to get KE you need to start the game and populate the board by rolling the dice? No two dice are allowed to display the same result at the same time. I suppose the dice should really be "thrown" by photons ...

If this article http://www.theguardian.com/science/life-and-physics/2012/feb/28/1 is correct, you do not need to consider this separately, atom by atom, as every electron in the universe needs its own energy level?

Best wishes.

PS I did recently buy a book on thermodynamics Thermodynamics by Enrico Fermi (1936). A Dover reprint. So maybe I am not too biassed against thermodynamics after all. However I have not read the book much as yet. I was shocked to find the book receipt showed August 2012, whereas I thought it would be summer 2014!

[Yablon]

Just last night I was able, I believe, to derive the laws of thermodynamics from equation (9.17) of http://vixra.org/pdf/1412.0267v1.pdf, by putting that into differential forms and then expanding it in the manner of the integral formulation of Maxwell's equations. If you try yourselves, keep in mind that dd=0, and notice that this is just like Maxwell's equations for monopoles, except one differential order higher, dd tau=0 versus ddA=0. This is part of how thermodynamics is seen to effectively "fill the gap" in electrodynamics, because the d_u tau parallels the d_u/\ in gauge theory, but the former is an observable gradient of an observable "thermal charge" which is the monopole residue, and the latter is an unobservable phase gradient. So the missing degree of freedom in the gauge field (A^u has at most three independent components rather than 4) is a gap filled by the gauge field. And whereas d_uA^u=0 is a gauge condition in EM theory, d_u epsilon^u <> 0 using the thermal vector epsilon^u turns into one of the thermodynamic laws.

I got stuck for awhile on a loop integral over time, then realized that that signified a reversible system. If you then change the time loop into an ordinary integral over time that cannot loop, you break the symmetry, establish the "arrow of time" (Hawking), have to change an equal sign to a greater than or equal sign, and bring about the thermodynamic second law of increasing entropy, all in one fell swoop. Traveling today for a funeral, but will see if I can do a quick summary writeup tonight. Jay