Vector potential of the Magnetic monopole

[andresB]

It is argued here that the use of vector potential to describe the magnetic field of a monopole is inherently wrong

http://arxiv.org/ftp/physics/papers/0701/0701232.pdf

It will indicate that the affirmation that charge quantization will be proved if a magnetic monopoles exists is wrong.

The argument seems convincing to me but I would like to her the opinion of more informed people about it..

 

[stedwards]

Interesting. I'll read it in the morning

 

[fzero]

According to the author (pg 10): "It does not seem correct to consider a non Euclidean geometry or recourse to bundle theory for the case of magnetic monopole as Wu and Yang have done." In fact, the mathematical techniques of both manifolds and fiber bundles are required to properly understand the singularity of Dirac's solution. To simply dismiss either is ignorant and unfortunate.

What one learns from the approach of Wu and Yang is that the Dirac string singularity simply reflects the fact that the 2-sphere is a curved manifold so that a single set of spherical polar coordinates cannot cover the whole space. In the standard form, the azimuthal angle $\phi$ is not defined on the $z$-axis where $\theta=0$ or $\pi$. These singularities are realized as the Dirac string in his solution.

Mathematicians learned to deal with this type of issue by recognizing that a given set of coordinates (a chart) would be well-defined on at least a small patch of the manifold. Then any manifold could be described given enough patches and corresponding charts. By comparing different charts in the areas where the patches overlapped, one learns the rules for changing coordinates from one patch to the next.

Wu and Yang showed that non-singular vector potentials can be defined on each of two patches on the sphere. On the overlap of patches, the vector potentials are related by a gauge transformation, so there is no physical difference in the choice. However, by using the two patches, the Dirac string singularity is gone and only the physical singularity at the origin (where the monopole is) remains.

So contrary to the opinion of the author, the Wu-Yang approach actually cures all of the physical problems with the original approach of Dirac.

A very readable description of Wu-Yang by Yang himself is http://physics.unm.edu/Courses/Finley/p495/handouts/CNYangonMonopoles.pdf .

 

[vanhees71]

I've read only the introductory section, and it's already utterly wrong. It's easy to show that there is not one map of the entire manifold $\mathbb{R}^3 \setminus \{(0,0,0)\}$, where the monopole is representable by a vector potential. But there's gauge invariance, and you can find a map with two charts, where you have a vector potential, and covering the entire manifold. In the overlap region of the charts the vector potentials deviate only by a gradient, i.e., they are connected by a gauge transformation. Since not the four-potential represents the electromagnetic field but the equivalence class of four-potentials given by a four-potential modulo a gauge transformation.

This point of view can be used to derive Dirac's charge-quantization condition from gauge invariance. So there's nothing wrong with Dirac's papers. For more details, see

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

 

[andresB]

I understand what fezero and vanshee71 are saying, that is the standard treatment of the magnetic monopole, unfortunately that don't entirely clarify the issue for me.

There are are a couple of things that bugs me

a) His use of the Helmholtz theorem to disqualify the use of a vector potential seem solid (one can't decompose a vector field in two different ways because the decomposition is unique) .

b) "It was shown that the Dirac vector potential for a magnetic monopole actually is the vector potential representing the field of a semi-infinite thin solenoid or magnet. This vector potential cannot represent the field of two different physical phenomena at the same time. The line of singularity for the solenoid (magnet) is physical and acceptable, but it is not for a point monopole having isotropic spherical symmetry."

 

[fzero]

I understand what fezero and vanshee71 are saying, that is the standard treatment of the magnetic monopole, unfortunately that don't entirely clarify the issue for me.

There are are a couple of things that bugs me

a) His use of the Helmholtz theorem to disqualify the use of a vector potential seem solid (one can't decompose a vector field in two different ways because the decomposition is unique) .

It is known that if magnetic charges are included, there is a duality in the Maxwell equations that allows us to swap electric quantities with magnetic quantities. Then, for static sources, the magnetic field can be written as the gradient of the dual scalar potential, while the electric field is given by the curl of a dual vector potential. In this dual formulation, the electric charges are topologically non-trivial constructions.

Recall that there is a physical significance to the assignment of potentials to a 4-vector, since, at least in QM, we must couple to the potential rather than the E and B fields. If we choose to express the magnetic field in terms of a scalar potential, we cannot also introduce a scalar potential for the electric field. Such a formalism is not self-consistent.

b) "It was shown that the Dirac vector potential for a magnetic monopole actually is the vector potential representing the field of a semi-infinite thin solenoid or magnet. This vector potential cannot represent the field of two different physical phenomena at the same time. The line of singularity for the solenoid (magnet) is physical and acceptable, but it is not for a point monopole having isotropic spherical symmetry."

As we've argued, the Wu-Yang construction shows that that the line of singularity in the Dirac solution is an artifact of the coordinate system. A more careful description of the solution uses at least two sets of coordinates and there is no line of singularity, only the physical singularity at the origin.