Potential formulation of Electrodynamics with magnetic sources

[cosmic dust]

Hello! I am trying to construct (if it is possible) a potential formulation of an electromagnetic theory which permits the presence of magnetic sources, using as a starting point the equations referred here:

http://en.wikipedia.org/wiki/Magnetic_monopole

Although I think that I have make some progress, I would like to know if there is such a theory already. Does anybody has something in mind?

If the answer is negative, I would also like to know if there is some particular reason why such a theory cannot not exist.

Thank's in advance!

 

[vanhees71]

Sure, as far as I know Dirac was the first to think about this issue:

P. A. M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60 (1931)
http://www.jstor.org/stable/95639

P. A. M. Dirac, Theory of Magnetic Poles, Phys. Rev. 74, 817 (1948), http://link.aps.org/abstract/PR/v74/i7/p817

One of my favorites about the subject is

T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12, 3845 (1975), http://link.aps.org/abstract/PRD/v12/i12/p3845

Of course you find this topic also covered in many good textbooks on electromagnetics like

J. D. Jackson, Classical Electrodynamics (at least in the 2nd edition it's covered)

J. Schwinger, Classical Electrodynamics

 

[mfb]

1) It is possible to construct a magnetic vector-potential based on moving electric charges
2) It is possible to construct an electric scalar potential based on electric charges

I think you can use (2) to add magnetic charges to the magnetic potential (giving it a 4th component) and (1) to add moving magnetic charges to the electric potential (giving it 3 additional components).

Edit: Oh, there are references.

 

[cosmic dust]

Sure, as far as I know Dirac was the first to think about this issue:

P. A. M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60 (1931)
http://www.jstor.org/stable/95639

P. A. M. Dirac, Theory of Magnetic Poles, Phys. Rev. 74, 817 (1948), http://link.aps.org/abstract/PR/v74/i7/p817

One of my favorites about the subject is

T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12, 3845 (1975), http://link.aps.org/abstract/PRD/v12/i12/p3845

Of course you find this topic also covered in many good textbooks on electromagnetics like

J. D. Jackson, Classical Electrodynamics (at least in the 2nd edition it's covered)

J. Schwinger, Classical Electrodynamics

Sorry, but I can't acces to the papers at the links. Can I find them somewhere else, for free?

And something else: as I deduced from the titles, these papers seemed to refer to the subject at quantum level. If correct, is there somehting in classical level?

 

[vanhees71]

The reason for the papers being about quantum physics is that the existence of magnetic monopoles would provide an explanation for the fact that the electrical charge in nature is "quantized". So far as we know all elementary particles (and the hadrons, which are the smallest lumps of strongly interacting matter that have been observed as free particles) come in integer multiples of the elementary charge e=\simeq 1.602 \cdot 10^{-19} \mathrm{C}.

We neither know a deeper reason for this specific value nor for a deeper explanation for the fact that there is such an elementary charge. The mathematical structure of both classical electromagnetics and quantum electrodynamics do not require such a atomistic nature of electric charge. The strong and weak forces as well as gravity are examples, where this is different. The coupling constants must be universal, because the non-abelian gauge structure of the underlying (quantum) field theories of the strong and the weak forces and the strong equivalence principle, which is the basic assumption leading to the General Theory of Relativity, describing gravity, would not work if these coupling constants weren't universal.

Also, there is no deeper reason known, why there shouldn't be magnetic monopoles in nature. Only empirical evidence tells us that there seem to be none. The interesting point is now that Dirac figured out that, if there were magnetic monopoles in nature, then electric charge would have to be quantized to keep the gauge invariance of the electromagnetic interaction, without which this very successful model for a large part of nature, would become inconsistent.

I'm not sure, how to get these papers other than going to a university library and download them from a computer connected to their network. There you also should find the textbooks (at least Jackson's book should be in any physics library).

 

[cosmic dust]

@Vanhess71

I understand that in order to explain the charge quantization, one has to assume the existence of magnetic monopoles. But my concern is not why should there be some magnetic charge. What I want to do is to axiomaticaly accept the existence of magnetic sources (i.e. the existence of a magnetic charge density and of a magnetic charge current), regardless if those charges are quantized or not. In other words I want to work in classical level, where charges can be distributed continuously in space.

As a staring point of this theory, I use the equations presented at the link I gave. So my question is, can these EM fields be represented by some potentials? Is there already a classical theory that does this? And if not, this is because someone has not dealt with the problem or there is some fundamental reason that these fields cannot be represented by potential?

This concern of mine comes from my curiosity to see if and how the gauge transformations altered with the presence of magnetic sources.

 

[vanhees71]

No, at the places where there is a magnetic charge and/or current density, there are no potentials anymore. The equations of motion in four-dimensional form are (in Heaviside-Lorentz units)
\partial_{\mu} F^{\mu \nu}=\frac{1}{c}j_e^{\nu}, \quad \partial_{\mu} (^{\dagger}F^{\mu \nu})=\frac{1}{c} j_m^{\nu},
where j_e^{\mu} and j_m^{\mu} are the electric and magnetic four-current. In usual electrodynamics the latter current is 0, and the equation are the homogeneous Maxwell equations, which can be read as integrability conditions for the existence of the electromagnetic four-potential.

The point, however, is that if you just have a single magnetic point monopole (analgous to a single electric point charge), then the integrability condition is only violated along the world line of the monopole. Everywhere else there exists a four-vector potential. This leads Dirac's description of the pole by four-vector potentials that are singular along a spacelike string, originating at the position of the pole at any given point. The shape of the string is quite arbitrary, and changing the shape of this line of singularities results only in a gauge transformation of the four-potential. This is the approach by Wu and Yang.

All this first can be formulated in terms of classical electrodynamics and its extension to the existence of a magnetic monopole. As I said, have a look at Jackson's textbook, where this topic is treated nicely.

 

[cosmic dust]

No, at the places where there is a magnetic charge and/or current density, there are no potentials anymore.

Would you mind if I decribe you my attempt to find some potentials? As it looks so far, I have managed to find some potentials that represent the fields (where the connection between fields and the potentials is more compicated then the usual one) and also I managed to find the gauge tranfrosmations for these potentials, which are sightly different than the usual ones.

The only problem that I have so far, is that the above have sense only if there are exist two functions, which are defined by some differential equations. Therefore it needs to be checked if these equations have solutions. Also, if these equations have some gauge arbitarines, then the potentials gauge tranformations should acquire an extra degree of freedom.

Or maybe the above are non-sence, because my way of thinking is wrong. So, would you mind to check it?