Solution Set to the Wu-Yang Differential Eqn. for Monopoles

[Yablon]

To all:

Here is my most recent work on the Dirac-Wu-Yang magnetic monopoles which explain the Fractional Quantum Hall Effect:

http://vixra.org/pdf/1503.0054v1.pdf

This responds in full to an earlier rejection received from PRD a few weeks ago, and has been resubmitted to that journal.

The introduction contains a very detailed review of the Dirac and 't Hooft papers, as well as detailed reasons why the approach I take here is correct and advocated by no less a figure than Dirac.

Section 2 has some solutions I had not seen or written up before, particularly the GUT solutions, and it clearly establishes a topological basis not only for charge quantization, but for energy quantization.

Best to all,

Jay

[Q-reeus]

Jay; while much of your differential forms presentation is outside (i.e. above) my level, I get that there is this flexibility to allow various and independent 'running' of mu (megnetic charge) and e (electric charge). Evidently synonymous with varying and independently variable levels of respective force carrier Boson massiveness (Yukawa-type fields)? But it's not clear to me whether your monopoles now are exclusively or optionally of the original Dirac type (ends of filamentary flux tubes), or 'dressed' 'point particles' thus a true dual of e as per ME's. Could you clarify that matter please? No doubt it is clarified in the paper but on my lazy and low-level read can't quite see where.

[Yablon]

Jay; while much of your differential forms presentation is outside (i.e. above) my level, I get that there is this flexibility to allow various and independent 'running' of mu (megnetic charge) and e (electric charge). Evidently synonymous with varying and independently variable levels of respective force carrier Boson massiveness (Yukawa-type fields)? But it's not clear to me whether your monopoles now are exclusively or optionally of the original Dirac type (ends of filamentary flux tubes), or 'dressed' 'point particles' thus a true dual of e as per ME's. Could you clarify that matter please? No doubt it is clarified in the paper but on my lazy and low-level read can't quite see where.

Q-reeus:

Let me take a stab at an answer which I hope will be satisfactory. I would have to say that these are monopoles of the original Dirac type, because they are derived from a differential equation which, as one of its solutions, gives the original DQC, and also because I am not using any type of spontaneous symmetry breaking in the manner of 't Hooft, which is what leads to real, observable, topologically stable monopoles which would be a true dual per Maxwell. However, because the original Dirac monopoles asymptotically behave identically to some of the 't Hooft monopole solutions (which depend on the GUT group G and how it is broken down), I would also say that the monopoles I am describing to correspond with the asymptotic behavior of some TP monopoles and so could be true dressed particles. There are several points in the paper which I refer to the asymptotic correspondence, because the point I was trying to bring out (whether I succeeded is another question) is that these could be physical monopoles via a connection to the analysis of 't Hooft. This also makes the GUT arena (eqs. (2.23) to (2.27) and related discussion) a bit tricky because there you presumably can no longer use the asymptotic solution but are getting right on top of the bare particle. In general, a connection to TP is one of the "physical interpretations" that one would engage in, after totally elaborating the mathematical solutions of the Wu-Yang differential equation. Dirac's two-step is a very good pedagogical approach to always bear in mind. Jay

[Q-reeus]

Q-reeus:
Let me take a stab at an answer which I hope will be satisfactory. I would have to say that these are monopoles of the original Dirac type, because they are derived from a differential equation which, as one of its solutions, gives the original DQC, and also because I am not using any type of spontaneous symmetry breaking in the manner of 't Hooft, which is what leads to real, observable, topologically stable monopoles which would be a true dual per Maxwell. However, because the original Dirac monopoles asymptotically behave identically to some of the 't Hooft monopole solutions (which depend on the GUT group G and how it is broken down), I would also say that the monopoles I am describing to correspond with the asymptotic behavior of some TP monopoles and so could be true dressed particles. There are several points in the paper which I refer to the asymptotic correspondence, because the point I was trying to bring out (whether I succeeded is another question) is that these could be physical monopoles via a connection to the analysis of 't Hooft. This also makes the GUT arena (eqs. (2.23) to (2.27) and related discussion) a bit tricky because there you presumably can no longer use the asymptotic solution but are getting right on top of the bare particle. In general, a connection to TP is one of the "physical interpretations" that one would engage in, after totally elaborating the mathematical solutions of the Wu-Yang differential equation. Dirac's two-step is a very good pedagogical approach to always bear in mind. Jay

Thanks Jay. It has sort of helped. It always seemed to me that original Dirac 'monopole' was somewhat fanciful in that there were imo big unaddressed questions over dynamical stability of the 'real entity' i.e. flux tube. Like what would prevent spontaneous longitudinal contraction, or transverse knotting/bulging/snaking etc. Maybe ultimately no more an issue than stability of a 'point particle' but I somehow doubt that.
It may not matter in the context of your scenarios envisaged but as I pointed out elsewhere a Dirac monopole would interact very differently with say a ring supercurrent than would a ME dual monopole. Point monopole vs Dirac 'monopole' free to circulate within an ordinary toroidal solenoid is another scenario that brings out issues (threading vs 'cutting' etc.). So imo 'identical dynamics' may have a somewhat limited range of validity but to repeat that may have no bearing on your application.

Anyway will try and watch in now and then to see how your project develops - especially feedback from peers who understand your ideas and maths language far better than I.

[Yablon]

Thanks Jay. It has sort of helped. It always seemed to me that original Dirac 'monopole' was somewhat fanciful in that there were imo big unaddressed questions over dynamical stability of the 'real entity' i.e. flux tube. Like what would prevent spontaneous longitudinal contraction, or transverse knotting/bulging/snaking etc. Maybe ultimately no more an issue than stability of a 'point particle' but I somehow doubt that.
It may not matter in the context of your scenarios envisaged but as I pointed out elsewhere a Dirac monopole would interact very differently with say a ring supercurrent than would a ME dual monopole. Point monopole vs Dirac 'monopole' free to circulate within an ordinary toroidal solenoid is another scenario that brings out issues (threading vs 'cutting' etc.). So imo 'identical dynamics' may have a somewhat limited range of validity but to repeat that may have no bearing on your application.

Anyway will try and watch in now and then to see how your project develops - especially feedback from peers who understand your ideas and maths language far better than I.

Q-rees: "Questions over dynamical stability of the 'real entity' i.e. flux tube" are very real questions, and those are the questions that 't Hooft and Polyakov first began to resolve which is why their work on this topic is important. As to the flux tubes, keep in mind that the Dirac flux tube is a completely fictitious artifact of the coordinate system, and is physically unobservable. It is a piece of mathematical scaffolding that we use to get a result and then are able to discard. You may wish to review the appendix of the paper, where I was surprised to have come upon the Dirac strings through an approach different from any I have seen before, based on the running of the charge strength. Jay

[Q-reeus]

Jay - In the static case, is the B field of yours or 't Hooft-Polyakovs monopole given by the curl of the usual magnetic vector potential A, or by -grad of a scalar magnetic potential? The analogous situation is Amperian current loop vs Gilbert 'true' magnetic dipole: https://en.wikipedia.org/wiki/Magnetic_potential
https://en.wikipedia.org/wiki/Magnetic_ ... he_H-field (and following subsection)