Covering Groups Permitting Monopoles with Fractional Charges

[Yablon]

Dear Friends:

I now have a new paper posted at: http://vixra.org/pdf/1504.0025v1.pdf. It is titled "Single-Valued Simply-Connected Covering Groups Permitting Well-Defined Dirac-Wu-Yang Monopoles with Fractional Charges." I have submitted this to PRD.

Briefly, on March 14 the paper at http://vixra.org/pdf/1503.0054v1.pdf which I submitted to PRD was rejected for the following reasons:

The Wu-Yang result is completely equivalent to Dirac's, with the only difference being that it is cast in the more modern language of fiber bundles. Making the Dirac string unobservable is equivalent to the requirement that the fiber bundle be well-defined. The bundle only satisfies this requirement if the standard Dirac condition is satisfied. Violating this condition necessarily leads to physically observable singularities and is thus unacceptable.

This new paper was in answer to this rejection. I also sent it with a cover letter quoted below:

I originally began to write this paper as an appendix to my paper DC11621 to demonstrate that your review of March 14 was wrong. But then, as I pursued this in detail, I came to conclude that you were right. But, only in the circumstance where the fractional mathematical solutions of the Wu and Yang differential equation are analyzed using Dirac strings in multivalued SO(3). Sections 2 and 3 of the attached, culminating in (3.8), derive in detail what you have been maintaining from the start as regards being restricted to only the standard DQC without fractions.

But of course there are single-valued simply-connected covering groups which can be homomorphically projected surjectively onto SO(3), and it occurred to me that the kernels for these groups can be developed using the Euler relation (4.1) for various roots of unity and that this had an identity with the Wu-Yang fractional charges. From there I developed the rest of the attached, and was able to show that when one uses the right covering groups, the fractional charges can indeed be projected onto SO(3) in a well-defined, unambiguous fashion.

One thing I have tried to determine but am uncertain about is whether the covering groups I have developed here have previously been developed elsewhere and just been given some different symbolic nomenclature. But I have not found this anywhere. And, if someone else has developed these groups before by a different name, I have found no evidence that they have been applied in any way to Dirac monopoles or to the question of fractional charges. If these have been developed before, I’d appreciate a suitable pointer.

Although I have been writing about this subject for several months, everything here is new, and is in line with views you have previously communicated. I appreciate your careful consideration of these results.

I am hopeful this will finally establish that these fractional charges can exist contrary to the main basis on which the paper has previously been rejected, so I can move on to having people recognize that my Fractional Quantum Hall results and the unifying connections I have proposed between thermodynamics and electrodynamics are also correct.

You will see why also why I now am very interested in fractional angle formulas.

Best to all,

Jay

[Ben6993]

Hi Jay

My eye caught this while browsing your new paper:

http://vixra.org/pdf/1504.0025v1.pdf page 15 lines 6 and 7:
"These features of the Clifford algebra in SO(1,3) are then projected out onto SO(1,3) ."

Did you mean to use SO(1,3) twice in that extract? Looks odd especially with using the word "out". Sometimes you refer to projecting onto SO(1,3) and sometimes onto SO(3).


And if SO wasn't difficult enough, you are using nth roots of SO ....
The nth roots of 1 are presumably different to the nth roots which lead to complex numbers (n=2), quaternions (n=4) and octonions (n=8)? Or are they the same but it is OK because you are not using them (i.e. where n does not equal 2, 4 or 8) as bases for algebras ... or...?
(If this sounds odd, just ignore it ... I am not an expert.)

[Yablon]

Hi Jay

My eye caught this while browsing your new paper:

http://vixra.org/pdf/1504.0025v1.pdf page 15 lines 6 and 7:
"These features of the Clifford algebra in SO(1,3) are then projected out onto SO(1,3) ."

Did you mean to use SO(1,3) twice in that extract? Looks odd especially with using the word "out". Sometimes you refer to projecting onto SO(1,3) and sometimes onto SO(3).

That was an error. Should be SL(2C).

And if SO wasn't difficult enough, you are using nth roots of SO ....
The nth roots of 1 are presumably different to the nth roots which lead to complex numbers (n=2), quaternions (n=4) and octonions (n=8)? Or are they the same but it is OK because you are not using them (i.e. where n does not equal 2, 4 or 8) as bases for algebras ... or...?
(If this sounds odd, just ignore it ... I am not an expert.)

Yup, the nth roots are pretty cool,IMHO. I am still trying to figure out for myself how to characterize them. I especially would like to use this to explain why the observed fractional FQHE charges are 2, then 3, 5, 7, 9, 11... all odd numbers. I think I can do that given the structural constraints on these nth root spinors (which I am inclined to call "fractors" or "rooters.") If I take an odd root spinor, e.g., at the m=1/3 root, I cannot multiply two of them to get an even denominator fractor. But I just though of this a half hour ago; will need to see if it pans out. After a Seder we are hosting with half the world.

Happy Passover / Easter to all,

Jay

[Yablon]

Dear Friends:
I now have a new paper posted at: http://vixra.org/pdf/1504.0025v1.pdf. It is titled "Single-Valued Simply-Connected Covering Groups Permitting Well-Defined Dirac-Wu-Yang Monopoles with Fractional Charges." I have submitted this to PRD. . .

This paper was rejected quickly, with the following brief rationale:

SU(2) is the simply connected universal covering group of SO(3). Despite your algebraic manipulations, SO(3) has no other covering groups.

I had the feeling that once I started talking about covering groups and the like I'd run into this sort of thing, because there is a whole language that has developed when it comes to Lie algebras and the misuse of the language can immediately cause a hangup on the other end for someone who may be better steeped in the language than I. I believe that the "algebraic manipulations" in my paper are correct and physically meaningful / observable, but I I need to tune up my use of this language.

Specifically: at http://en.wikipedia.org/wiki/Covering_s ... sal_covers it is stated that:

A covering space is a universal covering space if it is simply connected. The name universal cover comes from the following important property: if the mapping q: D → X is a universal cover of the space X and the mapping p : C → X is any cover of the space X where the covering space C is connected, then there exists a covering map f : D → C such that p ∘ f = q. This can be phrased as follows:

The universal cover (of the space X) covers any connected cover (of the space X).

I am actually glad to understand the special status that SU(2) has as a "universal cover" of SO(3) because after I wrote the paper I realized that I needed to find a special status for SU(2) over all the other roots groups to explain why the FQHE charges are all odd, except for the 1/2 unit of charge. Now I have that answer: A fractional charge state will not observed unless its generator is a primitive root of the unity matrix and the square of its generator is also a primitive root of the unitymatrix. That, together with SU(2) being a universal cover, means that the only fractional charge denominators will be 2,3,5,7,9..., just as is seen. But it will take more than this paragraph to explain how I get to this result. That will be in the next draft.

For the moment, referring to the Wiki excerpt, the more correct way to talk about my results in this paper is as follows:

The mapping q: SU(2) → SO(3) is a universal cover of the space SO(3). The higher root mappings I have used in my paper are , where the covering space is connected. This means that there exists a covering map such that p ∘ f = q. This covering map f is what I show in my equations (6.9) and (6.12).

Indeed, when I made the statement on page 24 prior to (6.13) that "for any higher root m>2 there will be an additional projection onto SO(3) which gets routed through SU(2)," this was my way of saying, perhaps with awkward language, that SU(2) is the universal covering group of SO(3). I will need to perhaps change how I discuss and symbolically represent this, but that does not alter the result.

Were we to talk about this using the kernels of these groups, we would say that is the universal cover kernel of 1, that are other kernel covers of 1, and that there is a covering map such that where . The left-superscript numbers outside the radicals represent fractions of a rotation through the unit circle in the complex Euler plane.

Jay