Reconciling Dirac Quantization Condition with no monopoles

[Yablon]

Dear Friends,

I would like to report a new paper which I have posted to http://vixra.org/pdf/1502.0083v1.pdf and also submitted to Physical Review D.

This paper continues my previous development of the Fractional Quantum Hall Effect (HQHE) but has a different approach and is much more consolidated and crisp. And it contains some results not previously reported.

Last month the two paper submissions reported here, one on FQHE and the other on thermodynamics, were rejected while I was out of the US on holiday. But the FQHE paper rejections were based either on grounds of "not here, send it to a condensed matter journal," and once I did that, "too long a paper, no review on the merits." And the thermodynamics paper was properly rejected on content: while the paper was correct that there is a set of Maxwell-type equations governing thermodynamics that start with a scalar potential rather than a vector potential A at the top differential order, I did not properly associate that scalar with the correct thermodynamic state variables in the correct form (which correct form is , where U is an internal energy field and T is the temperature also understood as a field). I have been working to develop this formally and fully, but that is not part of the present paper.

This present paper is focused on showing how the Dirac Quantization Condition (DQC) can be reconciled with the apparent absence of electric / magnetic duality in the material world. This should get me past the "send it to a condensed matter journal" rejections and if someone still tells me that, it is now compacted enough that I can do so and hopefully avoid the "paper is too long" rejections. All I report regarding FQHE is simply a consequence of completing the Dirac-Wu-Yang (DWY) derivation and eliminating the contradiction posed by the non-observation of monopoles.

What I have not reported in any previous papers which I now report here, is the fact that topological quantum number n=1,2,2,3,4,5... which represents charge quantization at low temperature where there are monopoles, turns into a energy quantization number once the monopoles dissolve and the electric / magnetic symmetry is broken at larger temperatures, see (3.19). I have also connected this charge-quantization-turned-energy-quantization number to the principal quantum number of atomic structure, see (3.21). Thus, taken with the orbital numbers I have previously reported and connected to orientation-entanglement-twist topology, this new paper provides a complete topological quantization for the entire Periodic Table of the Elements, in addition to its explanation of the FQHE. But I found (not without some struggle) a way to present this without a few dozen pages of discussion of the "bar and ribbon" apparatus that I used to help me originally lock into these results last autumn. And I have laid out very simply and concisely, several experimental predictions which I believe will confirm the results in this paper.

Nice to be back online.

Jay

[FrediFizzx]

Hi Jay,

I bet you are wishing that you stayed in Mexico with another polar vortex headed your way?

I have been having trouble reconciling the DQC with regular electrodynamics. In regular electrodynamics, magnetic charge is simply electric charge times the speed of light. g = q*c using g for magnetic charge and q for electric charge. In CGS units we can set the electric and magnetic force equations equal,

q^2/r^2 = g^2/(c^2 r^2).

So we can see that g = q*c. And you can also find this to be true in SI or other unit systems. So with n = 1, I don't see how to get to the DQC q*g = 2pi in the unit system you are using of hbar = c = eps0 = 1?

[Yablon]

...So with n = 1, I don't see how to get to the DQC q*g = 2pi in the unit system you are using of hbar = c = eps0 = 1?

Hi Fred,

With the fundamental constants restored, the DQC for n=1 is . If you are having trouble with length-time-mass dimensionality, think of two Planck masses put into the numerator of Newton's law with a gravitational coupling G. Those masses are defined as . So , i.e. it has the same dimensionality as mass time the square root of a coupling constant; same with .

Jay

[FrediFizzx]

Hi Jay,

That doesn't really help. In the limit of hbar --> 0, we should be able to recover the classical electrodynamic result of magnetic charge being electric charge times the speed of light. Do you happen to have Dirac's original derivation for the DQC that you could email me? Thanks.

[Ben6993]

Hi Fred

hbar is a constant http://en.wikipedia.org/wiki/Planck_constant
How can it tend to nought?
Am I missing something?

[FrediFizzx]

In classical electrodynamics, there is no hbar. I probably should have written, "in the classical limit of hbar --> 0".

[Yablon]

Hi Jay,

That doesn't really help. In the limit of hbar --> 0, we should be able to recover the classical electrodynamic result of magnetic charge being electric charge times the speed of light. Do you happen to have Dirac's original derivation for the DQC that you could email me? Thanks.

Hi Fred:

Actually, Dirac's paper is posted online at http://rspa.royalsocietypublishing.org/ ... l.pdf+html. As it happens, I have been very closely re-studying this paper over the last week and am going to add a new section to my paper which very carefully relates Dirac's original work to where I have moved it in the present day. I should have something to share early to middle of this coming week.

I tend to think of the classical limit as the rather than limit of , but the main result either way is that you then get (no monopoles) and e remains undetermined. You will see from Dirac that the undetermined nature of e is a major facet (and in his words, disappointment) of his paper. He started out trying to explain the fine structure number 1/137.036 and instead all he got was the lousy t-shirt, er, I mean, this DQC. To this day people are still at it on the 137 number.

Jay

PS: If you look at surveys of the top theoretical physicists of the 20th century, of course Einstein is in the #1 slot, but #2 can be very contested. I absolutely put Dirac in that position, ahead of Bohr and Heisenberg and other usual suspects. And what he did with monopoles is a significant part of that because I believe that the road to theoretical unification is paved with monopoles and he started that.

[FrediFizzx]

Thanks Jay,

OK, problem solved. Mu is not magnetic charge; it is magnetic flux in the CGS units that Dirac is using. And that is what he says it is in the paper on page 68. It just so happens that in CGS units magnetic flux and electric charge have the same dimensions. Easy to see since hbar*c = (electric charge)^2 so hbar*c/e = electric charge. Or... in this case magnetic flux.

[FrediFizzx]

So... the DQC does not ruin what I originally thought about magnetic charge. A photon sees electric charge as magnetic charge since g = q*c. Magnetism is a relativistic effect. Now... how does this relate to the DQC?

[Yablon]

So... the DQC does not ruin what I originally thought about magnetic charge. A photon sees electric charge as magnetic charge since g = q*c. Magnetism is a relativistic effect. Now... how does this relate to the DQC?

Fred,

I would think of the DQC as a nonrelativistic form of magnetism which emerges out of the fact that charge is quantized. Keep in mind, Maxwell's equations contain no net magnetic flux through any closed surface. But neither do they require that charge be quantized.so the quantization of charge goes hand-in-hand with the existence of monopoles, if you take the DQC strictly at face value. My own work refines that view somewhat, because under all conditions where the DQC is not actually observed, which is to say when the temperature is not near absolute zero, the charge quantum number of the DQC turns into the principal quantum number for electrons in atomic shells and goes from representing charge quantization to representing energy quantization.

Jay

[FrediFizzx]

Hi Jay,

Putting magnetic charge in terms of the magnetic flux quantum we have,



So I think the DQC is relativistic as can be seen when we go to the fine structure constant in the formula. And even though the magnetic flux quantum seems to be stronger in relation to a quantum of electic charge, magnetic charge quantum is equal to electric charge quantum when
.

[FrediFizzx]

Of course in our non-relativistic world, since magnetic charge is electric charge cranked up by the speed of light, then maybe Dirac was right. Magnetic poles are too hard to separate.

[Yablon]

Hi Jay,

Putting magnetic charge in terms of the magnetic flux quantum we have,



So I think the DQC is relativistic as can be seen when we go to the fine structure constant in the formula. And even though the magnetic flux quantum seems to be stronger in relation to a quantum of electic charge, magnetic charge quantum is equal to electric charge quantum when

Hi Fred,

This is how I would do your calculation:

Start with the DQC in SI units:

(1)

Divide through to isolate on the right, which puts everything into units of magnetic flux (Weber) denoted :

(2)

Multiply through by to obtain:

(3)

The squared electric charge is related to the dimensionless fine structure number (1/137.036 at low probe) by:

(4)

Use this in (3) to finally obtain:

(5)

This gives us your which is a magnetic flux times the electric charge strength. I am not sure of your interpretation of this in relation to all the way on the right. The main thing I read out of (5) after dividing out is:

(5)

which tells me that the magnetic flux is an integer times the speed of light.

Jay

[FrediFizzx]

Hi Jay,

That can't be right. Magnetic flux in CGS units has the same dimensions as electric charge, cm^1.5 gm^0.5 sec^-1. Your mistake is in eq. (2). is magnetic flux. It should be,



Which we can see the units are right since hbar*c is charge^2 divided by charge gives units of charge.

[Yablon]

Hi Jay,

That can't be right. Magnetic flux in CGS units has the same dimensions as electric charge, cm^1.5 gm^0.5 sec^-1. Your mistake is in eq. (2). is magnetic flux. It should be,



Which we can see the units are right since hbar*c is charge^2 divided by charge gives units of charge.

Fred,

I agree something is amiss. Go to http://en.wikipedia.org/wiki/Magnetic_m ... antization, middle box, SI Weber. That is what I used for flux because it removes the from the denominator as you did with your derivation. There must be something hidden in that which I do not see. This is why I just stick with natural units all the time to avoid the arcane details of different systems of units.

Jay

[FrediFizzx]

Hi Jay,

I don't ever use that SI units (weber convention). Don't know what it is really. I have my MathCad program setup for regular SI units or Gaussian-CGS units as they are the most common for physics besides natural units of or The last one is the natural units you use in your paper since you specify or in your natural units

So in your natural units magnetic flux is,



Magnetic charge is,

This is wrong; see below.
.

[Yablon]

... Magnetic charge is,

.

Fred,

The bottom line calculation I get for with and included is:



I have been scratching my head as to how you equate this to magnetic charge.

Jay

[FrediFizzx]

Hi Jay,

Oops sorry, I think I put the n in the wrong place on the LHS so it is messed up.

Ok, referring to Dirac's paper on page 68, he shows for magnetic flux,



He is using Gaussian-CGS units where not your Heaviside-Lorentz (H-L, another commonly used unit system) units of I showed earlier in this thread that magnetic charge is g = ec in CGS units. So rearranging the above equation,



Then multiply both sides by e,



Gives us,

in CGS units or in CGS natural units

So now to convert to your H-L units. g is still equal to ec in H-L units so going thru the same procedure as we did for CGS units above,



Then multiply thru by n/c,

In your H-L units which you have correct.

Fred

[Yablon]

... I showed earlier in this thread that magnetic charge is g = ec in CGS units.

Fred,

You did say that, but please explain or provide a reference as to why . (Magnetic) charge = (electric) flux times velocity (of light). That is what I am not seeing or not familiar with.

Is this somehow related to the field strength bivector:

?

If so, how?

I also found a neat 6x6 table at http://en.wikipedia.org/wiki/Dimensiona ... anck_units, but I do not see (Magnetic) charge = (electric) flux times velocity (of light) there.

Jay

[FrediFizzx]

Hi Jay,

Well, in classical electrodynamics it is actually g = q*c but with q = e, the electric charge quantum, we should get the hypothetical magnetic charge quantum. There is one thing we are certain of in CE is that k_e/k_m = c^2. IOW, the electric constant divided by the magnetic constant is equal to c^2. So for the force of electric charge quantum we have,



and for the force between hypothetical magnetic charge quanta,



Now in H-L units, k_e = 1/4pi and k_m = k_e/c^2 = 1/4pi c^2. So make those replacements and set the two equations equal and you get,

-->

I suppose a question here would be why are the two force equations equal? No particular reason; we are forcing them to be equal so that g will be in terms of e and c. It has nothing at all to do with fields.

Fred