4.4.2 Dirac Monopole#

Prompts

What is a magnetic monopole, and why don’t we observe them in nature? How would their existence change electromagnetism?
Dirac showed that monopoles must come with a singularity—a “string”—in the vector potential. Why is this string unavoidable, and why don’t we observe it?
State Dirac’s quantization condition and explain what it tells us about the relationship between electric and magnetic charge. What does it imply for charge quantization?
A charged particle in a monopole background must have a single-valued wavefunction. How does this constraint force the existence of the Dirac string?
Is the Dirac string real or just a mathematical artifact? What experimental consequences would distinguish between these interpretations?

Lecture Notes#

Overview#

What if magnetic charges existed? A point magnetic monopole would produce a radial \(\boldsymbol{B}=g\hat{\boldsymbol r}/(4\pi r^2)\), restoring the electric-magnetic duality of Maxwell’s equations. But \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\) implies \(\nabla \cdot \boldsymbol{B} = 0\), creating a contradiction. Dirac resolved this by allowing the vector potential to be singular along a semi-infinite line — the Dirac string. The string is unphysical if the Aharonov-Bohm phase around it is trivial, which forces the Dirac quantization condition \(qg = 2\pi n\hbar\). A remarkable consequence: if even one monopole exists anywhere in the universe, all electric charges must be quantized. The two-patch formulation reveals the topological origin: the winding number of the gauge transition function on the equator equals the monopole charge, classified by \(\pi_1(U(1)) = \mathbb{Z}\).

The Concept of a Magnetic Monopole#

For the monopole discussion we use the standard Dirac-monopole convention where magnetic charge is defined directly by flux. Maxwell’s equations without magnetic charge are:

\[\begin{split} \begin{split} \nabla \cdot \boldsymbol{E} &= \frac{\rho_e}{\epsilon_0},\\ \nabla \times \boldsymbol{E} &= -\frac{\partial \boldsymbol{B}}{\partial t},\\ \nabla \cdot \boldsymbol{B} &= 0,\\ \nabla \times \boldsymbol{B} &= \mu_0\left(\boldsymbol{j}_e + \epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}\right). \end{split} \end{split}\]

The equation \(\nabla \cdot \boldsymbol{B} = 0\) reflects that no magnetic charge has ever been observed. However, if magnetic monopoles existed with charge density \(\rho_m\), this equation would become:

\[ \nabla \cdot \boldsymbol{B} = \rho_m\]

and the equations would point toward an electric-magnetic duality between electric and magnetic sources.

A point magnetic monopole at the origin with charge \(g\) produces the field:

\[ \boldsymbol{B}(\boldsymbol{r}) = \frac{g}{4\pi}\,\frac{\hat{\boldsymbol{r}}}{r^2}\]

This is the magnetic analogue of the Coulomb electric field of a point charge \(q\):

\[ \boldsymbol{E}(\boldsymbol{r}) = \frac{q}{4\pi\epsilon_0}\,\frac{\hat{\boldsymbol{r}}}{r^2}\]

Why Monopoles Are Conceptually Natural

Even though no monopoles have been observed, they appear naturally in:

Quantum electrodynamics as topological solitons
Grand unified theories (GUTs) as endpoints of symmetry breaking
Gauge theories on non-trivial topological manifolds
Condensed matter systems (magnetic monopoles in spin ice)

Their non-observation does not mean they are unphysical—only that they may be massive or very rare.

The Problem: Vector Potential Singularity#

In electromagnetism with monopoles, we face a fundamental problem. The magnetic field satisfies:

\[ \nabla \cdot \boldsymbol{B} = g\,\delta^3(\boldsymbol{r}) \neq 0\]

everywhere on the sphere far from the origin. Therefore, we cannot write \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\) with a smooth vector potential everywhere on \(\mathbb{R}^3 \setminus \{\text{origin}\}\).

This is a topological obstruction. The first Chern number of the monopole field is non-zero, preventing a globally-defined vector potential.

The Dirac String Solution#

Dirac’s ingenious solution is to allow the vector potential to be singular along a line (the Dirac string) extending from the monopole to infinity. Away from the string, \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\) holds exactly.

Consider the north-patch potential (string along the negative \(z\)-axis, valid for \(\theta < \pi\)):

\[ \boldsymbol{A}^{(N)}(\boldsymbol{r}) = \frac{g}{4\pi r} \frac{1 - \cos\theta}{\sin\theta}\hat{\boldsymbol{\varphi}}\]

In spherical coordinates, the curl gives:

\[ \nabla \times \boldsymbol{A}^{(N)} = \frac{g}{4\pi}\,\frac{\hat{\boldsymbol{r}}}{r^2} \quad \text{for } \theta < \pi\]

This correctly reproduces the monopole field away from the string. At \(\theta = \pi\) (the string), the potential diverges, but the observable magnetic field remains well-defined (at least in the physical sense).

String Location is Gauge-Dependent

The position of the Dirac string is not physical—it is a gauge artifact. Different gauges place the string in different locations (e.g., pointing along \(+z\), or along the \(x\)-axis). All physical observables must be independent of string location.

This is why the string is sometimes called a “gauge singularity” rather than a physical object.

Aharonov-Bohm Phase and Quantization#

In quantum mechanics, a charged particle moving through a region with a non-trivial gauge field acquires a topological phase:

\[ \Phi_{\mathrm{AB}} = \frac{q}{\hbar}\oint_{\mathcal{C}} \boldsymbol{A}\cdot\mathrm{d}\boldsymbol{l}. \]

The two-patch construction makes the consequence transparent. The north-patch potential \(\boldsymbol{A}^{(N)}\) has a Dirac string at \(\theta = \pi\); the south-patch potential \(\boldsymbol{A}^{(S)}\) has its string at \(\theta = 0\). On the equatorial overlap, the two differ by a gauge transformation \(\boldsymbol{A}^{(N)} - \boldsymbol{A}^{(S)} = \nabla\alpha\) with \(\alpha(\varphi)\) winding around the equator. The wavefunctions on the two patches are related by the local phase

\[ \psi^{(N)} = \mathrm{e}^{\mathrm{i}q\alpha(\varphi)/\hbar}\,\psi^{(S)}. \]

For both wavefunctions to be single-valued going once around the equator (\(\varphi:0\to 2\pi\)), the phase factor \(\mathrm{e}^{\mathrm{i}q\alpha/\hbar}\) must return to itself — i.e., the winding of \(\alpha\) must satisfy

\[ \frac{q}{\hbar}\bigl[\alpha(2\pi) - \alpha(0)\bigr] = 2\pi n,\qquad n \in \mathbb{Z}. \]

The winding of \(\alpha\) around the equator equals the total magnetic flux of the monopole through any closed surface enclosing it. To bridge from the abstract winding to the geometric flux, integrate \(\boldsymbol{A}^{(N)} - \boldsymbol{A}^{(S)} = \nabla\alpha\) around the equator and apply Stokes’ theorem to each patch — \(\boldsymbol{A}^{(N)}\) is smooth on the upper hemisphere (whose outward normal induces the \(+\hat{\boldsymbol{\varphi}}\) orientation on the equator), \(\boldsymbol{A}^{(S)}\) is smooth on the lower hemisphere (whose outward normal induces \(-\hat{\boldsymbol{\varphi}}\), so \(-\oint_{\mathrm{eq}}\boldsymbol{A}^{(S)}\cdot\mathrm{d}\boldsymbol{l} = +\int_{\mathrm{lower}}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S}\)):

\[\begin{split} \begin{split} \alpha(2\pi) - \alpha(0) &= \oint_{\mathrm{eq}}\nabla\alpha\cdot\mathrm{d}\boldsymbol{l} = \oint_{\mathrm{eq}}\boldsymbol{A}^{(N)}\cdot\mathrm{d}\boldsymbol{l} - \oint_{\mathrm{eq}}\boldsymbol{A}^{(S)}\cdot\mathrm{d}\boldsymbol{l} \\ &= \int_{\mathrm{upper}}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S} + \int_{\mathrm{lower}}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S} = \oint_{S^{2}}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S} \\ &= \int\frac{g}{4\pi r^{2}}\,r^{2}\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi = g. \end{split} \end{split}\]

Each hemisphere contributes \(g/2\) to the total flux; together they reconstruct the full sphere integral. Substituting and solving for \(qg\) delivers the quantization condition:

Dirac quantization condition

(175)#\[ qg \;=\; 2\pi n\hbar,\qquad n \in \mathbb{Z}. \]

The full monopole flux is \(g\). Demanding that the Aharonov-Bohm phase \(qg/\hbar\) be a multiple of \(2\pi\) gives the same condition.

Derivation: same condition from the single-patch loop integral

A direct line-integral on the equator \(\theta = \pi/2\) using \(\boldsymbol{A}^{(N)}\) yields, with \(\mathrm{d}\boldsymbol{l} = r\sin\theta\,\mathrm{d}\varphi\,\hat{\boldsymbol{\varphi}}\) on the latitude circle of radius \(r\sin\theta\),

\[\begin{split} \begin{split} \oint_{\mathcal{C}}\boldsymbol{A}^{(N)}\cdot\mathrm{d}\boldsymbol{l} &= \int_{0}^{2\pi}\underbrace{\frac{g}{4\pi r}\frac{1-\cos\theta}{\sin\theta}}_{A^{(N)}_\varphi}\;\underbrace{r\sin\theta\,\mathrm{d}\varphi}_{|\mathrm{d}\boldsymbol{l}|}\bigg|_{\theta=\pi/2} \\ &= \int_{0}^{2\pi}\frac{g}{4\pi}\bigl(1-\cos(\pi/2)\bigr)\,\mathrm{d}\varphi = \frac{g}{2}. \end{split} \end{split}\]

This counts only the flux through the upper hemisphere (\(g/2\), half the total), so it gives a phase \(\Phi_{\mathrm{AB}}^{(N)} = qg/(2\hbar)\) — apparently demanding \(qg = 4\pi n\hbar\) if used naively.

Repeating with the south-patch potential gives a phase of opposite sign: \(\Phi_{\mathrm{AB}}^{(S)} = -qg/(2\hbar)\). Both phases are valid in their respective patches; their difference on the overlap is what must be a multiple of \(2\pi\):

\[ \Phi_{\mathrm{AB}}^{(N)} - \Phi_{\mathrm{AB}}^{(S)} = \frac{qg}{\hbar} = 2\pi n \quad\Longrightarrow\quad qg = 2\pi n\hbar. \]

The factor of \(2\) is therefore unavoidable; the single-patch calculation alone misses it because it only sees half the monopole flux.

Charge Quantization from Monopoles#

The most profound consequence of Dirac’s result is a prediction of electric charge quantization.

If even one magnetic monopole exists anywhere in the universe with charge \(g\), then the existence of the wavefunction forces:

\[ e g = 2\pi n_e\hbar\]

for the electron charge \(e\). Simultaneously, for any other particle with charge \(q\):

\[ qg = 2\pi n_q\hbar\]

Taking the ratio:

\[ \frac{q}{e} = \frac{n_q}{n_e}\]

This is a rational number. If all charges in nature are rational multiples of some fundamental unit, charge is quantized.

Observation: All observed elementary particle charges are indeed quantized in units of \(e/3\) (the basic quark charge). This is naturally explained by monopole quantization, even though monopoles have never been detected.

Discussion: Monopole Charge Quantization

Dirac quantization requires \(e g = 2\pi n\hbar\) in the convention used here.

If a single magnetic monopole exists somewhere in the universe, what does this constraint imply about the quantization of electric charge? Is it a prediction of monopole existence, or merely a consistency requirement once one is assumed? And conversely, could a future cosmological observation — for example, the absence of monopole signatures in any astrophysical environment — falsify Dirac’s argument, or only constrain the abundance and mass of monopoles?

Two-Patch Gauge Formulation#

Since no single gauge covers the sphere without a singularity, we use two coordinate patches (Northern and Southern hemispheres) with a gauge transformation relating them on the overlap.

North patch (\(\theta < \pi/2 + \epsilon\)): String at south pole

\[ \boldsymbol{A}^{(N)} = \frac{g}{4\pi r}\frac{1 - \cos\theta}{\sin\theta}\hat{\boldsymbol{\varphi}}\]

South patch (\(\theta > \pi/2 - \epsilon\)): String at north pole

\[ \boldsymbol{A}^{(S)} = -\frac{g}{4\pi r}\frac{1 + \cos\theta}{\sin\theta}\hat{\boldsymbol{\varphi}}\]

In the overlap region, the two potentials differ by a gauge transformation:

\[ \boldsymbol{A}^{(N)} - \boldsymbol{A}^{(S)} = \nabla\alpha\]

where:

\[ \alpha(\varphi) = \frac{g}{2\pi}\varphi\]

The wavefunction in the overlap must satisfy:

\[ \psi^{(N)} = \mathrm{e}^{\mathrm{i}q\alpha/\hbar}\psi^{(S)}\]

Note that \(\alpha(\varphi + 2\pi) - \alpha(\varphi) = g\). The charged wavefunction sees the phase \(\exp[\mathrm{i}q\alpha/\hbar]\), so single-valuedness again requires \(qg/\hbar=2\pi n\).

Topological Origin of Quantization

The quantization of magnetic charge arises from the topology of the fiber bundle over the 2-sphere \(S^2\). The structure group is \(U(1)\) (the phase symmetry), and a charged particle in the monopole background sees a non-trivial \(U(1)\) bundle whose transition function has winding number \(n=qg/h\).

Single-valuedness of the transition function forces \(n\in\mathbb{Z}\), hence Dirac quantization.

Poll: Monopole charge quantization

A magnetic monopole with charge \(g\) produces \(\boldsymbol{B}=g\hat{\boldsymbol r}/(4\pi r^2)\). Dirac showed that if monopoles exist, the product of electric and magnetic charge obeys \(qg=2\pi n\hbar\). Which physical principle most precisely captures the origin of this quantization condition?

(A) Gauge invariance of the quantum-mechanical phase under continuous gauge transformations of the vector potential.

(B) The discreteness of the vector potential circulation \(\oint \boldsymbol{A} \cdot \mathrm{d}\ell = \pi \hbar / q\).

(C) The single-valuedness of the electron wavefunction (equivalently, the patch-overlap transition function) on a manifold with a monopole singularity.

Summary#

A magnetic monopole produces a radial \(\boldsymbol{B}\) field; no globally smooth vector potential exists (topological obstruction)
Dirac’s resolution: allow \(\boldsymbol{A}\) to be singular along a string; the string is unphysical if the AB phase around it is \(2\pi n\)
The Dirac quantization condition \(qg = 2\pi n\hbar\) follows from single-valuedness of the wavefunction
If one monopole exists, all electric charges are quantized as rational multiples of a fundamental unit
The two-patch formulation reveals the topological origin: monopole charge = winding number of the \(U(1)\) transition function on \(S^2\)
Monopole vs Maxwell: a point magnetic charge would source \(\boldsymbol{B} = g\hat{\boldsymbol r}/(4\pi r^2)\) away from the Dirac string; standard Maxwell with no magnetic charge enforces \(\nabla\cdot\boldsymbol{B} = 0\), so such an isolated \(1/r^2\) monopole Coulomb field is excluded.

Homework#

1. Monopole field. In the flux-defined (Dirac-monopole) convention used throughout this lesson, a magnetic monopole of charge \(g\) at the origin produces \(\boldsymbol{B} = g\hat{\boldsymbol r}/(4\pi r^2)\).

(a) Compute \(\nabla \cdot \boldsymbol{B}\) for \(r \neq 0\) and show it vanishes. What happens at \(r = 0\)?

(b) Compute the total magnetic flux through a sphere of radius \(R\) centered on the monopole. Express the result in terms of \(g\).

(c) Compare this monopole field to the electric field of a point charge \(q\). Write down the duality transformation that maps one to the other.

2. String singularity. The vector potential \(\boldsymbol{A}^{(N)} = \frac{g}{4\pi r}\frac{1 - \cos\theta}{\sin\theta}\hat{\boldsymbol{\varphi}}\) has a Dirac string along \(\theta = \pi\).

(a) Verify that \(\nabla \times \boldsymbol{A}^{(N)} = \frac{g}{4\pi r^2}\hat{\boldsymbol r}\) for \(\theta \neq \pi\).

(b) Show that \(\boldsymbol{A}^{(N)}\) diverges as \(\theta \to \pi\). This is the Dirac string — explain why it is a gauge artifact, not a physical singularity.

(c) Write the south-patch potential \(\boldsymbol{A}^{(S)}\) with string along \(\theta = 0\). Find the gauge function \(\alpha(\varphi)\) such that \(\boldsymbol{A}^{(N)} - \boldsymbol{A}^{(S)} = \nabla\alpha\).

3. Apply Dirac quantization. The Dirac quantization condition \(qg = 2\pi n\hbar\) relates electric charge \(q\) and magnetic charge \(g\).

(a) For an electron (elementary charge \(e\), taken positive in the body convention of § Charge Quantization from Monopoles) and the smallest possible monopole (\(n = 1\)), compute \(g_0 = h/e\) and write its SI value with units.

(b) For a Cooper pair (\(q = -2e\)), find the smallest allowed monopole charge \(g_0'\) and compare to the electron case.

(c) Suppose a particle with charge \(q = e/3\) existed as a free particle. What would the smallest allowed monopole charge be? Why does the existence of fractional charge constrain the monopole spectrum more tightly than the electron alone?

(d) Conclude: if a single monopole were ever observed, its charge would directly bound the smallest electric charge in the universe. State this implication precisely.

4. Charge quantization. If a monopole with minimum charge \(g_0 = h/e\) exists, all electric charges must be quantized.

(a) Show that any particle with charge \(q\) must satisfy \(q = ne\) for integer \(n\).

(b) Quarks have charges \(\pm e/3\) and \(\pm 2e/3\). What minimum monopole charge \(g_0'\) is consistent with quark charges?

(c) No monopole has been observed despite extensive searches. Propose one alternative explanation for charge quantization that does not require monopoles.

5. Two-patch gauge formulation. In the overlap region between north and south patches, the gauge transition function is \(U_{NS}(\varphi) = \mathrm{e}^{\mathrm{i}n\varphi}\) where \(n = qg/h\).

(a) Single-valuedness of \(U_{NS}\) requires \(n \in \mathbb{Z}\). Show this reproduces the Dirac quantization condition.

(b) The integer \(n\) is the winding number of the map \(U_{NS}: S^1 \to U(1)\). Explain why two maps with different winding numbers cannot be continuously deformed into each other.

6. Electric-magnetic duality. Write Maxwell equations with both electric and magnetic sources, then discuss the formal symmetry that exchanges electric and magnetic sectors.

(a) Explain why the unscaled replacement \(\boldsymbol E\to\boldsymbol B\) is not dimensionally meaningful in SI units. Then describe qualitatively how the modified Maxwell equations with both electric and magnetic sources become more symmetric after field rescaling.

(b) Write the Lorentz force on a dyon (a particle carrying both electric charge \(q\) and magnetic charge \(g\)) and verify it is duality-covariant.

(c) In quantum mechanics, \(\boldsymbol{B} = \nabla \times \boldsymbol{A}\) requires \(\nabla \cdot \boldsymbol{B} = 0\). Explain why this breaks the duality and makes monopoles nontrivial.

7. Angular momentum in monopole field. A charged particle near a monopole has modified angular momentum \(\hat{\boldsymbol{J}} = \hat{\boldsymbol{r}} \times \hat{\boldsymbol{p}} + s\hat{\boldsymbol{r}}\) with \(s = qg/(4\pi\hbar)\).

(a) Explain why \(\hat{\boldsymbol{L}} = \hat{\boldsymbol{r}} \times \hat{\boldsymbol{p}}\) alone does not satisfy the standard SU(2) algebra in a monopole background.

(b) Verify that \([\hat{J}_i, \hat{J}_j] = \mathrm{i}\hbar\epsilon_{ijk}\hat{J}_k\) is restored after adding \(s\hat{\boldsymbol{r}}\).

8. Monopole detection. Grand Unified Theories predict monopoles with mass \(\sim 10^{16}\) GeV/\(c^2\).

(a) A monopole passing through a superconducting loop of inductance \(L = 1\,\mu\)H induces a current jump \(\Delta I = \Phi_0'/L\). Estimate \(\Delta I\) using \(\Phi_0' = h/(2e)\).

(b) The Parker bound limits monopole flux to \(F < 10^{-15}\,\text{cm}^{-2}\,\text{sr}^{-1}\,\text{s}^{-1}\). How many monopoles would pass through a \(1\,\text{m}^2\) detector per year?

(c) Explain why the enormous mass of GUT monopoles makes them extremely difficult to produce in accelerators, and why cosmological searches focus on relic monopoles from the early universe.