Magnetic monopoles and Noether theorem

[tom.stoer]

Is there a Lagrangian from which the modified Maxwell equations including a magnetic charge density (magnetic monopoles) can be derived?
Can one introduce a matter part (like in the Dirac Lagrangian) which reproduces the magnetic charge density?
Does this Lagrangian have a symmetry which corresponds to conserved magnetic charged? How does this symmetry look like?

 

[sheaf]

Interesting question!

I guess your requirement of having fermionic sources for the magnetic charge rules out obtaining monopoles in the usual symmetry breaking fashion (i.e. cheating by doing fancy things with the gauge fields at infinity).

I was wondering what happens if you "sacrifice" the Maxwell equations $$\partial_{\mu}*F^{\mu\nu}=0$$ so that instead of having zero on the RHS you have some magnetic source. But if you do that, then you've removed the requirement that $F$ is a closed two form, so you can no longer use the vector potential $A^{\mu}$ in the normal way. You then lose all the machinery of gauge covariant derivatives etc. So I'm not sure how to proceed to look for a theory with this symmetry.

Maybe there's some cleverly supersymmetric way of obtaining such a thing, but I can't see it...

 

[tom.stoer]

I guess your requirement of having fermionic sources for the magnetic charge rules out obtaining monopoles in the usual symmetry breaking fashion (i.e. cheating by doing fancy things with the gauge fields at infinity).

Yes, that was my intention.

I was wondering what happens if you "sacrifice" the Maxwell equations $$\partial_{\mu}*F^{\mu\nu}=0$$ so that instead of having zero on the RHS you have some magnetic source. But if you do that, then you've removed the requirement that $F$ is a closed two form, so you can no longer use the vector potential $A^{\mu}$ in the normal way. You then lose all the machinery of gauge covariant derivatives etc. So I'm not sure how to proceed to look for a theory with this symmetry.

Yes, that was my problem as well. I think it's no longer possible to construct something like the Dirac Lagrangian for the minimal coupling. So the new theory is not something like "Dirac + new terms". At this time I decided to ask this question here in he forum.

Perhaps one could do something like introducing a non-trivial spacetime topology.

 

[TrickyDicky]

Perhaps one could do something like introducing a non-trivial spacetime topology.

I don't think anything other than the euclidean topology is compatible with the physics observed.

 

[torsten]

Yes, the magnetic monopole is strongly related to a non-trivial topology or better to a non-trivial connection (where one needs at least two charts). The approach went back to Wu and Yang around 1975.
Then a singularity-free description can be constructed, if we give up the traditional parametrization of the space $\mathbb{R}^3$ surrounding the monopole, by a single set of coordinates. Instead let us divide $\mathbb{R}^3\setminus 0$ into two slightly overlapping hemispheres, say the north hemisphere RN and the south one RS. Then $RN\cap RS$ is the ''equator''. Then, one can write two potentials AN and AS, which are singularity-free everywhere in the domains of their definition:
$AN=g\frac{1 − cos\theta}{r\cdot sin\theta}e_\phi$ for $0\leq\theta<\pi/2+\epsilon$ in RN
$AN=g\frac{1 + cos\theta}{r\cdot sin\theta}e_\phi$ for $\pi/2-\epsilon<\theta\leq\pi$ in RS
in the intersection region $RS\cap RN$ there is a gauge transformation to connect them
$AS\rightarrow AS-\frac{i}{e}exp(-2ieg\phi)\nabla exp(2ieg\phi)$
So, the magnetic monopole is connected with a non-trivial U(1) fiber bundle (defining a non-trivial connection)

 

[PAllen]

Is there a Lagrangian from which the modified Maxwell equations including a magnetic charge density (magnetic monopoles) can be derived?
Can one introduce a matter part (like in the Dirac Lagrangian) which reproduces the magnetic charge density?
Does this Lagrangian have a symmetry which corresponds to conserved magnetic charged? How does this symmetry look like?

This seems close to what you are looking for:

http://arxiv.org/abs/math-ph/0203043

 

[fzero]

This seems close to what you are looking for:

http://arxiv.org/abs/math-ph/0203043

Well this doesn't specify the matter apart from supposing there is some current. It's actually worse than that, since the theory cannot be quantized. If you try to quantize a theory with two gauge potentials and a single gauge group, you'll find that the gauge symmetry can be used to remove the negative norm state from only one of the potentials.

I don't believe that the formalism Tom is seeking actually exists. This is both for the reasons that sheaf pointed out, but also because monopolar field configurations are nonperturbative, so we wouldn't expect to see fundamental sources for them in the same Lagrangian as the electric degrees of freedom. The closest thing to a Lagrangian description of monopoles is the one obtained via dualities in certain cases. An older review of the subject is Harvey.

 

[PAllen]

Well this doesn't specify the matter apart from supposing there is some current. It's actually worse than that, since the theory cannot be quantized. If you try to quantize a theory with two gauge potentials and a single gauge group, you'll find that the gauge symmetry can be used to remove the negative norm state from only one of the potentials.

I didn't read the OP as asking for a theory that could be quantized. The paper I reference clearly states it is classical.

 

[torsten]

The situaton for monopoles is the same as for instantons. For an instanon, there is also no Lagrangian. The field equation is given by the duality relation F=*F (the global extremum of the Yang-Mills action).
Monopoles have its root in the same formalism. The Hodge star changes from electric to magnetic currents and vice versa. The basic symmetry is therefore the modular group $SL(2,\mathbb{Z})$.
But one important point is necessary: one needs a non-trivial topology, i.e. there is a non-contractable loop (or a non-trivial element of the funamental group).

The paper (as reference of PAllen) used a trick to extend the Maxwell tensor by using two U(1) fields. But maybe Tom has exactly this in mind.

 

[tom.stoer]

@PAllen: I have to double check, but it seems that this is exactly what I was looking for

@Torsten: I am aware of the topological considerations and the self-duality, but I was really looking for a Lagrangian, simply b/c I want to understand if it's possible to derive both dF=j and d*F=k from a Lagrangian; the reason is that this extension of the Maxwell equations looks symmetric at first glance, but it isn't b/c for the magnetic current you need some additional (topological) tricks;

@fzero: all I had in mind was an extension of classical electrodynamics; quantization may be a second step

 

[samalkhaiat]

Is there a Lagrangian from which the modified Maxwell equations including a magnetic charge density (magnetic monopoles) can be derived?
Can one introduce a matter part (like in the Dirac Lagrangian) which reproduces the magnetic charge density?
Does this Lagrangian have a symmetry which corresponds to conserved magnetic charged? How does this symmetry look like?

I would like to show you the math when I have the time. And yes, your question was answered as far back as 1968 by Zwanziger. His local Lagrangian was invariant under $U_{e}(1) \times U_{m}(1)$ gauge group. See
D. Zwanziger, Phys. Rev. 176, 1489(1968).
D. Zwanziger, Phys. Rev. D3, 880(1971).
D. Zwanziger, Phys. Rev. D19, 1153(1979).

Also, it is worth looking at the non-local Hamiltonian formulation of Schwinger:
J. Schwinger, Phys. Rev. 144, 1087(1966).

Topological implications are disscussed in
P. Goddard and D. Olive, Rep. Prog. Phys. 41, 1357(1978).

Sam

 

[sheaf]

I would like to show you the math when I have the time. And yes, your question was answered as far back as 1968 by Zwanziger. His local Lagrangian was invariant under $U_{e}(1) \times U_{m}(1)$ gauge group. See
D. Zwanziger, Phys. Rev. 176, 1489(1968).
D. Zwanziger, Phys. Rev. D3, 880(1971).
D. Zwanziger, Phys. Rev. D19, 1153(1979).

Also, it is worth looking at the non-local Hamiltonian formulation of Schwinger:
J. Schwinger, Phys. Rev. 144, 1087(1966).

Topological implications are disscussed in
P. Goddard and D. Olive, Rep. Prog. Phys. 41, 1357(1978).

Sam

I'd be very interested in a brief explanation of this when you have the time (the Zwanziger references aren't easily/freely available). In particular how to think of the degrees of freedom implied by the existence of the "extra" U(1) connection.

 

[Trifis]

I would be more than interested in a continuation of this discussion.

Monopoles symmetrizing the Maxwell equations seem to cause a hell of a lot of problems as far as spacetime and Lagrangian formulation of electromagnetism is considered. We practically have to invent a new, highly non-trivial QED, new feynman rules etc!

Moreover, I cannot see how electricity and magneticism are to be considered completely symmetrical if the monopole sources of the first are just normal fermions, wheres the monopole sources of the latter are complicated topological configurations.