Yablon wrote: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:
PRD rejection wrote: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:
Wiki on universal covers wrote: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
\to SO(3))
, where the covering space
)
is connected. This means that there exists a covering map
\to {}_{m}^{n} \tilde{G}(2))
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
}]{1})
. The left-superscript numbers outside the radicals represent fractions of a rotation through the unit circle in the complex Euler plane.
Jay
