Dirac monopole

A solution to the Maxwell equations describing a point source of a magnetic field. In 1931, P.A.M. Dirac [a1] considered the quantum mechanics of the electron in a magnetic field (due to a point source),

\begin{equation} \tag{a1} \mathbf{B} = g \frac { \mathbf{r} } { r^{3} }, \end{equation}

where $r = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the length of the position vector $\mathbf{r} = ( x , y , z )$ in the Cartesian coordinates and $g$ is a constant determining the strength of the field, known as a magnetic charge of the monopole. Since the induction vector $\mathbf{B}$ in (a1) is central, it can be conveniently written in the spherical coordinates $r , \theta , \phi$ defined by $x = r \operatorname { sin } \theta \operatorname { cos } \phi$, $y = r \operatorname { sin } \theta \operatorname { sin } \phi$, $z = r \operatorname { cos } \theta$, $0 \leq \theta \leq \pi$, $0 \leq \phi < 2 \pi$. In these coordinates, only the radial component of $\mathbf{B}$ is non-zero and equals $B _ { r } = g / r ^ { 2 }$. Maxwell's equations imply that there is no single vector potential corresponding to $\mathbf{B}$ defined on the whole of $\mathbf{R} ^ { 3 }$. However, Dirac found that $\mathbf B = \nabla \times \mathbf A ^ { \pm }$, with vector potentials $\mathbf{A} ^ { + }$ whose only non-zero components are in the azimuthal direction and read

\begin{equation} \tag{a2} A _ { \phi } ^ { \pm } = \frac { g } { r \operatorname { sin } \theta } ( \pm 1 - \operatorname { cos } \theta ). \end{equation}

The potentials $\mathbf{A} ^ { + }$, $\mathbf{A}^{ - }$ are singular at $\theta = \pi$ (the negative $z$-axis) and $\theta = 0$ (the positive $z$-axis), respectively. These singularities are known as Dirac's string singularities. The union of the regions in which $\mathbf{A} ^ { + }$ are well-defined covers the whole of $\mathbf{R} ^ { 3 }$. In the intersection of these regions ($0 < \theta < \pi$) the vector potentials $\mathbf{A} ^ { + }$ are related by the gauge transformation, $A ^ { + } = A ^ { - } + \nabla \chi$, with $\chi = 2 g \phi$. If there is an electron in the magnetic field $\mathbf{B}$, then in the region where both $\mathbf{A} ^ { + }$ and $\mathbf{A}^{ - }$ are well-defined, the wave functions of the electron corresponding to different vector potentials should be related by the gauge transformation $\chi$, i.e.,

\begin{equation*} \Psi _ { + } = e ^ { i e \chi / \hbar } \Psi _ { - } = e ^ { 2 i e g \phi / \hbar } \Psi _ { - }, \end{equation*}

where $e$ is the electric charge of the electron and $\hbar$ is the Planck constant divided by $2 \pi$. The wave function $\Psi _ { + }$ is single valued if and only if $2 e g / \hbar = n$ for an integer $n$, i.e. if and only if the magnetic charge attains discrete values

\begin{equation} \tag{a3} g = n \frac { \hbar } { 2 e } , \quad n = 0 , \pm 1 , \pm 2 , \ldots . \end{equation}

Thus, the consistency of the monopole field (a1) with quantum mechanics can be achieved, provided the magnetic charge $g$ be quantized. Equation (a3) expresses also "duality" (reciprocity) between magnetic and electric charges: If $g$ and $e$ are interchanged, (a3) remains the same. Dirac used this fact to explain the observed quantization of the electric charge: Should a magnetic monopole of charge, say, $g$ exist, then by the above argument the electric charge would be allowed to have only discrete values $e = n \hbar / 2 g$. This argument, however, would leave the quantization of magnetic charge unexplained, a fact that Dirac found disappointing [a1].

In 1975, T.T. Wu and C.N. Yang [a9] observed that Dirac's monopole of magnetic charge $g = n \hbar / 2 e$ has a natural topological interpretation as a connection in the $U ( 1 )$ principal bundle over the two-sphere $S ^ { 2 }$ with the first Chern number (the winding number) $-n$ (cf. Connections on a manifold; Principal fibre bundle; or [a3] for a review). In natural units $\hbar = e = 1$, the potentials $\mathbf{A} ^ { + }$ can be written as one-forms

\begin{equation*} A ^ { \pm } = \frac { n } { 2 } ( \pm 1 - \operatorname { cos } \theta ) d \phi , \end{equation*}

and they are a connection one-form written in two charts covering $S ^ { 2 }$. More precisely, $\phi$, $\theta$ above are coordinates of the two-sphere. Then $\theta = 0$ is the north pole and $A ^ { - }$ is well-defined everywhere outside the north pole, for example on a chart $H_-$ covering the southern hemisphere including the equator ($\theta > \pi / 2 - \epsilon$). On the other hand, $\theta = \pi$ is the south pole, and thus $A ^ { + }$ is well-defined everywhere except the south pole, for example on a chart $H _ { + }$ covering the northern hemisphere including the equator ($\theta < \pi / 2 + \epsilon$). The intersection $H _ { + } \cap H _ { - }$ is parametrized by the azimuthal angle $\phi$. In order to combine this local system into a $U ( 1 )$-principal bundle, on $H _ { + } \cap H _ { - }$ the $U ( 1 )$-coordinate $\psi _ { + }$ over $H _ { + }$ must be related to the $U ( 1 )$-coordinate $\psi _ { - }$ over $H_-$ by $\psi + = \psi _ { - } - n \phi$, with integer $n$. This explains the appearance of Dirac's string singularity when the $A^{\mp}$ are extended to $H _ { \pm }$, and the fact that it can be removed by a gauge transformation which requires Dirac's quantization condition. Thus, the trivial bundle $S ^ { 2 } \times U ( 1 )$ admits no monopole (charge $0$-monopole). The existence of a monopole indicates non-triviality of a corresponding principal bundle. The monopole of charge $\hbar \nmid 2 e$ is the connection in the Hopf fibration $S ^ { 3 } \rightarrow S ^ { 2 }$, while the monopole of charge with $n > 1$ corresponds to the $U ( 1 )$-bundle over $S ^ { 2 }$ with the lens space $L _ { n } = \operatorname {SU} ( 2 ) / {\bf Z} _ { n }$ as a total space ($\mathbf{Z} _ { n }$ is viewed inside $\operatorname{SU} ( 2 )$ as a subgroup of $n$th roots of the unit matrix) [a7].

The Dirac monopole is an example of an Abelian monopole, i.e., a solution of field equations of gauge theory with Abelian gauge group $U ( 1 )$. Since the mid-1970{}s there has been a considerable interest in non-Abelian monopoles, in particular those related to the $\operatorname{SU} ( 2 )$ gauge theories. In pure mathematics this was triggered in particular by the appearance of $\operatorname{SU} ( 2 )$ gauge theory in the classification of four-manifolds by S.K. Donaldson [a2]. However, in 1994, E. Witten [a8] showed that certain Abelian monopole equations motivated by the supersymmetric quantum field theory [a5], [a6] and known as the Seiberg–Witten equations, can be used to derive both the Donaldson invariants of four-manifolds as well as new ones (the Seiberg–Witten invariants; cf. also Four-dimensional manifold). It was soon noted [a4] that the Dirac gauge potential $A ^ { - }$ with $n = - 1$ provides a bosonic part of the simplest (not $L^{2}$) solution to Seiberg–Witten equations. Witten's observation, as well as the appearance of magnetic monopoles in string theory, revived the interest in both monopoles and the reciprocity between electric and magnetic charges (electric-magnetic duality).

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