1.9 Magnetic monopoles

Maxwell’s equations unify electricity and magnetism. In the history of physics they should be recognized as the first attempt to unify forces in Nature. In spite of their great success, Dirac (1931) noticed that there existed an asymmetry in Maxwell’s equations: the equation $\mathrm{div}B=0$ denies the existence of magnetic charges. He introduced the magnetic monopole, a point magnetic charge, to make the theory symmetric.

1.9.1 Dirac monopole

Consider a monopole of strength $g$ sitting at $r=0$ ,

$$\mathrm{div}\mathbf{B}=4\pi g{\delta }^3(\mathbf{r}).\text{\hspace{1em}(1.275)}$$

It follows from $\Delta (1/r)=-4\pi {\delta }^3(r)$ and $\nabla (1/r)=-r/r^3$ that the solution of this equation is

$$\mathbf{B}=g\mathbf{r}/r^3.\text{\hspace{1em}(1.276)}$$

The magnetic flux $\Phi$ is obtained by integrating $B$ over a sphere $S$ of radius $R$ so that

$$\Phi ={\oint }_S\mathbf{B}\cdot \mathrm{d}\mathbf{S}=4\pi g.\text{\hspace{1em}(1.277)}$$

What about the vector potential which gives the monopole field (1.276)? If we define the vector potential $A^{\mathrm{N}}$ by

$$A_x^{\mathrm{N}}=\frac{-gy}{r(r+z)}{A}_y^{\mathrm{N}}=\frac{gx}{r(r+z)}{A}_z^{\mathrm{N}}=0\text{\hspace{1em}(1.278a)}$$

we easily verify that

$$\mathrm{curl}{\mathbf{A}}^{\mathrm{N}}=g\mathbf{r}/r^3+4\pi g\delta (x)\delta (y)\theta (-z).\text{\hspace{1em}(1.279)}$$

We have $\mathrm{curl}{A}^{\mathrm{N}}=B$ except along the negative $z$ -axis ( $\theta =\pi$ ). The singularity along the $z$ -axis is called the Dirac string and reflects the poor choice of the coordinate system. If, instead, we define another vector potential

$$A_x^S=\frac{gy}{r(r-z)}{A}_y^S=\frac{-gx}{r(r-z)}{A}_z^S=0\text{\hspace{1em}(1.278b)}$$

we have $\mathrm{curl}{A}^{\mathrm{S}}=B$ except along the positive $z$ -axis ( $\theta =0$ ) this time. The existence of a singularity is a natural consequence of (1.277). If there were a vector $A$ such that $B=\mathrm{curl}A$ with no singularity, we would have, from Gauss’ law,

$$\Phi ={\oint }_S\mathbf{B}\cdot \mathrm{d}\mathbf{S}={\oint }_S\mathrm{curl}\mathbf{A}\cdot \mathrm{d}\mathbf{S}={\int }_V\mathrm{div}(\mathrm{curl}\mathbf{A})\mathrm{d}V=0$$

where $V$ is the volume inside the surface $S$ . This problem is avoided only when we abandon the use of a single vector potential.

Exercise 1.14

Let us introduce the polar coordinates $(r,\theta ,\varphi )$ . Show that the vector potentials $A^{\mathrm{N}}$ and $A^{\mathrm{S}}$ are expressed as

$${\mathbf{A}}^{\mathrm{N}}(\mathbf{r})=\frac{g(1-\cos \theta )}{r\sin \theta }{\hat{\mathbf{e}}}_{\varphi }\text{\hspace{1em}(1.280a)}$$

$${\mathbf{A}}^S(\mathbf{r})=-\frac{g(1+\cos \theta )}{r\sin \theta }{\hat{\mathbf{e}}}_{\varphi }\text{\hspace{1em}(1.280b)}$$

where ${\hat{e}}_{\varphi }=-\sin \varphi {\hat{e}}_x+\cos \varphi {\hat{e}}_y$

1.9.2 The Wu-Yang monopole

Wu and Yang (1975) noticed that the geometrical and topological structures behind the Dirac monopole are best described by fibre bundles. In chapters 9 and 10, we give an account of the Dirac monopole in terms of fibre bundles and their connections. Here we outline the idea of Wu and Yang without introducing the fibre bundle. Wu and Yang noted that we may employ more than one vector potential to describe a monopole. For example, we may avoid singularities if we adopt $A^{\mathrm{N}}$ in the northern hemisphere and $A^{\mathrm{S}}$ in the southern hemisphere of the sphere $S$ surrounding the monopole. These vector potentials yield the magnetic field $B=gr/r^3$ , which is non-singular everywhere on the sphere. On the equator of the sphere, which is the boundary between the northern and southern hemispheres, $A^{\mathrm{N}}$ and $A^{\mathrm{S}}$ are related by the gauge transformation, $A^{\mathrm{N}}-A^{\mathrm{S}}=\mathrm{grad}\Lambda$ . To compute this quantity $\Lambda$ , we employ the result of exercise 1.14,

$${\mathbf{A}}^{\mathrm{N}}-{\mathbf{A}}^{\mathrm{S}}=\frac{2g}{r\sin \theta }{\hat{\mathbf{e}}}_{\varphi }=\mathrm{grad}(2g\varphi )\text{\hspace{1em}(1.281)}$$

where use has been made of the expression

$$\mathrm{grad}f=\frac{\partial f}{\partial r}{\hat{\mathbf{e}}}_r+\frac{1}{r}\frac{\partial f}{\partial \theta }{\hat{\mathbf{e}}}_{\theta }+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }{\hat{\mathbf{e}}}_{\varphi }.$$

Accordingly, the gauge transformation function connecting $A^{\mathrm{N}}$ and $A^{\mathrm{S}}$ is

$$\Lambda =2g\varphi .\text{\hspace{1em}(1.282)}$$

Note that $\Lambda$ is ill defined at $\theta =0$ and $\theta =\pi$ . Since we perform the gauge transformation only at $\theta =\pi /2$ , these singularities do not show up in our analysis. The total flux is

$$\Phi ={\oint }_S\mathrm{curl}\mathbf{A}\cdot \mathrm{d}\mathbf{S}={\int }_{U_{\mathrm{N}}}\mathrm{curl}{\mathbf{A}}^{\mathrm{N}}\cdot \mathrm{d}\mathbf{S}+{\int }_{U_{\mathrm{S}}}\mathrm{curl}{\mathbf{A}}^{\mathrm{S}}\cdot \mathrm{d}\mathbf{S}\text{\hspace{1em}(1.283)}$$

where $U_{\mathrm{N}}$ and $U_{\mathrm{S}}$ stand for the northern and southern hemispheres respectively. Stokes’ theorem yields

$$\begin{array}{rl}\Phi & ={\oint }_{\text{e q u a t o r}}{A}^{\mathrm{N}}\cdot \mathrm{d}s-{\oint }_{\text{e q u a t o r}}{A}^{\mathrm{S}}\cdot \mathrm{d}s={\oint }_{\text{e q u a t o r}}\left(A^{\mathrm{N}}-A^{\mathrm{S}}\right)\cdot \mathrm{d}s\\ & ={\oint }_{\text{e q u a t o r}}\mathrm{grad}(2g\varphi )\cdot \mathrm{d}s=4g\pi \end{array}\text{\hspace{1em}(1.284)}$$

in agreement with (1.277).

1.9.3 Charge quantization

Consider a point particle with electric charge $e$ and mass $m$ moving in the field of a magnetic monopole of charge $g$ . If the monopole is heavy enough, the

Schrodinger equation of the particle takes the form

$$\frac{1}{2m}{\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)}^2\psi (\mathbf{r})=E\psi (\mathbf{r}).\text{\hspace{1em}(1.285)}$$

It is easy to show that under the gauge transformation $A\to A+\mathrm{grad}\Lambda$ , the wavefunction changes as $\psi \to \exp (ie\Lambda /\hslash c)\psi$ . In the present case, $A^{\mathrm{N}}$ and $A^{\mathrm{S}}$ differ only by the gauge transformation $A^{\mathrm{N}}-A^{\mathrm{S}}=\mathrm{grad}(2g\varphi )$ . If ${\psi }^{\mathrm{N}}$ and ${\psi }^{\mathrm{S}}$ are wavefunctions defined on $U_{\mathrm{N}}$ and $U_{\mathrm{S}}$ respectively, they are related by the phase change

$${\psi }^{\mathrm{S}}(\mathbf{r})=\exp \left(\frac{-\mathrm{i}e\Lambda }{\hslash c}\right){\psi }^{\mathrm{N}}(\mathbf{r}).\text{\hspace{1em}(1.286)}$$

Let us take $\theta =\pi /2$ and study the behaviour of wavefunctions as we go round the equator of the sphere from $\varphi =0$ to $\varphi =2\pi$ . The wavefunction is required to be single valued, hence (1.286) forces us to take

$$\frac{2eg}{\hslash c}=nn\in \mathbb{Z}.\text{\hspace{1em}(1.287)}$$

This is the celebrated Dirac quantization condition for the magnetic charge; if the magnetic monopole exists, the magnetic charge takes discrete values,

$$g=\frac{\hslash cn}{2e}n\in \mathbb{Z}.\text{\hspace{1em}(1.288)}$$

By the same token, if there exists a magnetic monopole somewhere in the universe, all the electric charges are quantized.

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正文：英文 · OCR 机器稿 · 待校对

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