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 \(\boldsymbol{B} = (g/r^2)\hat{\boldsymbol{r}}\), 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 \(qg = n\hbar c/2\). 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#

Maxwell’s equations in SI units are:

\[ \nabla \cdot \boldsymbol{E} = \frac{\rho_e}{\epsilon_0}, \quad \nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}\]

\[ \nabla \cdot \boldsymbol{B} = 0, \quad \nabla \times \boldsymbol{B} = \mu_0\boldsymbol{j}_e + \mu_0\epsilon_0\frac{\partial \boldsymbol{E}}{\partial t}\]

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:

\[ \nabla \cdot \boldsymbol{B} = \mu_0\rho_m\]

and the equations would possess a beautiful electric-magnetic duality:

\[ \boldsymbol{E} \to \boldsymbol{B}, \quad \boldsymbol{B} \to -\boldsymbol{E}, \quad \rho_e \to \rho_m, \quad \boldsymbol{j}_e \to \boldsymbol{j}_m\]

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

\[ \boldsymbol{B}(\boldsymbol{r}) = \frac{\mu_0 g}{4\pi}\frac{\hat{r}}{r^2}\]

(In Gaussian units, the \(\mu_0/(4\pi)\) factor is replaced by 1.)

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

\[ \boldsymbol{E}(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\frac{q\hat{r}}{r^2}\]

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:

\[ \boldsymbol{\nabla} \cdot \boldsymbol{B} = \mu_0 g \delta^3(\boldsymbol{r}) \neq 0\]

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 \(\theta < \pi\)):

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

In spherical coordinates, the curl gives:

\[ \nabla \times \boldsymbol{A}^{(N)} = \frac{\mu_0 g}{4\pi r^2}\hat{r} \quad \text{for } \theta < \pi\]

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:

\[ \phi = \frac{q}{\hbar c}\oint_\mathcal{C} \boldsymbol{A} \cdot d\boldsymbol{l}\]

(In SI units, this is \(\phi = \frac{q}{\hbar}\oint \boldsymbol{A} \cdot d\boldsymbol{l}\); we use natural units with \(\hbar = c = 1\) for simplicity.)

Consider a closed loop \(\mathcal{C}\) that encircles the monopole (e.g., at the equator \(\theta = \pi/2\), going once around in \(\phi\)). Computing the line integral using the north-patch potential:

\[ \oint_{\mathcal{C}} \boldsymbol{A}^{(N)} \cdot d\boldsymbol{l} = \int_0^{2\pi} \frac{g}{r} \frac{1 - \cos(\pi/2)}{\sin(\pi/2)} \cdot r\sin(\pi/2) \, \mathrm{d}\phi = \int_0^{2\pi} g \, \mathrm{d}\phi = 2\pi g\]

So the phase picked up is:

\[ \phi = \frac{2\pi qg}{\hbar c}\]

Derivation: Dirac Quantization Condition

For the wavefunction to be single-valued after encircling the monopole once, the phase must be a multiple of \(2\pi\):

\[ \phi = 2\pi n, \quad n \in \mathbb{Z}\]

This requires:

\[ \frac{2\pi qg}{\hbar c} = 2\pi n\]

Simplifying:

\[ qg = n\hbar c, \quad n \in \mathbb{Z}\]

This is Dirac’s quantization condition. In Gaussian-CGS units (more common in QFT literature), it is written:

\[ qg = n\frac{\hbar c}{2}\]

The factor of 2 differs because of unit conventions.

Dirac’s quantization condition:

\[ qg = n\hbar c \quad \text{(SI with } \hbar = c = 1)\]

or in Gaussian units:

\[ qg = n\frac{\hbar c}{2}\]

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:

\[ eg = n_e\hbar c\]

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

\[ qg = n_q\hbar c\]

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 eg = nℏc/2.

Calculate the minimum magnetic charge g_min in SI units.

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 or just consistency?

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

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

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

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

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

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

where:

\[ \chi(\phi) = \frac{2g}{\hbar c}\phi\]

The wavefunction in the overlap must satisfy:

\[ \psi^{(N)} = \mathrm{e}^{-\mathrm{i}\chi}\psi^{(S)}\]

Note that \(\chi(\phi + 2\pi) - \chi(\phi) = \frac{4\pi g}{\hbar c}\). For the wavefunction to be globally single-valued (i.e., \(\psi^{(N)}(\phi + 2\pi) = \psi^{(N)}(\phi)\)), this winding number must be an integer, leading again to the quantization condition.

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 monopole of charge \(g\) corresponds to a non-trivial \(U(1)\) bundle with first Chern number \(c_1 = g/(\hbar c)\).

Single-valuedness of the wavefunction forces \(c_1 \in \mathbb{Z}\), hence quantization.

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 \(qg = n\hbar c/2\) 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 \(S^2\)

Aspect	With Monopole	Without Monopole
Magnetic field	\(\boldsymbol{B} = \frac{g\hat{r}}{4\pi r^2}\)	\(\boldsymbol{B} = 0\) at infinity
Vector potential	Singular string required	Smooth everywhere
Aharonov-Bohm phase	\(\phi = \frac{2\pi qg}{\hbar c}\)	\(\phi = 0\)
Charge quantization	\(qg = n\hbar c\) (forced)	\(q\) can be continuous
Physical observables	Independent of string location	Independent of gauge

Homework#

1. Monopole Field and Duality

Suppose a magnetic monopole with charge \(g\) is placed at the origin. Write down the magnetic field \(\boldsymbol{B}(\boldsymbol{r})\) and compare it to the electric field of a point electric charge \(q\) at the origin.

(a) Show that the monopole field obeys \(\nabla \cdot \boldsymbol{B} = \rho_m\), where \(\rho_m = g\delta^3(\boldsymbol{r})\) is the “magnetic charge density.”

(b) Under the duality transformation \(\boldsymbol{E} \to \boldsymbol{B}\), \(\boldsymbol{B} \to -\boldsymbol{E}\), \(e \to g\), \(g \to -e\), what do Maxwell’s equations become? Verify that they remain unchanged.

2. Vector Potential Singularity

The vector potential for a monopole with string along the negative \(z\)-axis is given in spherical coordinates as:

\[ \boldsymbol{A} = \frac{g}{r} \frac{1 - \cos\theta}{\sin\theta}\hat{\phi}\]

(a) Show that \(\nabla \times \boldsymbol{A} = \frac{g}{r^2}\hat{r}\) away from the string (i.e., for \(\theta \neq \pi\)).

(b) What happens to \(\boldsymbol{A}\) as \(\theta \to \pi\) (approaching the string)? Explain why this singularity is unphysical by describing what is physical (the magnetic field).

(c) Suppose you perform a large gauge transformation \(\boldsymbol{A} \to \boldsymbol{A}' = \boldsymbol{A} - \nabla\chi\) with \(\chi = \frac{2g}{\hbar c}\phi\). Show that this moves the string singularity to a different location on the sphere. Conclude that the string location is gauge-dependent.

3. Aharonov-Bohm Phase and Quantization

A charged particle with charge \(q\) travels in a closed loop \(\mathcal{C}\) that encircles a magnetic monopole at the origin. The particle acquires a topological phase:

\[ \phi = \frac{q}{\hbar c}\oint_\mathcal{C} \boldsymbol{A} \cdot d\boldsymbol{l}\]

(a) For a loop at constant \(\theta = \pi/2\) (equator), encircling the monopole once, calculate the line integral \(\oint \boldsymbol{A} \cdot d\boldsymbol{l}\) and show that:

\[ \phi = \frac{2\pi qg}{\hbar c}\]

(b) Explain why requiring the wavefunction to be single-valued after encircling the monopole leads to the quantization condition \(\phi = 2\pi n\).

4. Charge Quantization and Fundamental Scales

If even one magnetic monopole exists anywhere in the universe, then all electric charges must be quantized. Assume the existence of a monopole with minimal magnetic charge (the quantum of magnetic charge).

(a) If the electron has charge \(q = -e\) and a monopole has minimal charge \(g_0 = \frac{\hbar c}{2e}\), what is the product \(eg_0\)?

(b) For the wavefunction to be single-valued, what integer values of \(n\) are allowed in \(qg = n\frac{\hbar c}{2}\)? (Consider \(q = e, 2e, 3e, \ldots\))

(c) Explain how the observation that all elementary particles have charges \(0, \pm e, \pm 2e/3, \pm e/3, \ldots\) (quark and lepton charges) is consistent with monopole quantization, even if monopoles have never been found.

5. Two-Patch Gauge Formulation

Since no single vector potential covers the entire sphere, we use two coordinate patches (north and south) with a gauge transformation between them.

(a) Write down the north-patch potential \(\boldsymbol{A}^{(N)}\) and south-patch potential \(\boldsymbol{A}^{(S)}\). At what locations do these potentials singular?

(b) In the overlap region (e.g., \(\pi/4 < \theta < 3\pi/4\)), the two potentials differ by a gauge gradient: \(\boldsymbol{A}^{(N)} - \boldsymbol{A}^{(S)} = \nabla\chi\). Determine \(\chi(\phi)\) and show that it has winding number 2 around the \(z\)-axis (i.e., \(\chi(\phi + 2\pi) - \chi(\phi) = 4\pi g/(\hbar c)\)).

\[ \psi^{(N)}|_{\text{overlap}} = \mathrm{e}^{-\mathrm{i}\chi}\psi^{(S)}|_{\text{overlap}}\]

Explain why single-valuedness of the full wavefunction requires the winding number to be quantized, and how this leads to Dirac’s quantization condition.

6. Aharonov-Bohm Geometry (Conceptual)

Consider a charged particle constrained to move on a sphere (radius \(R\)) centered at a monopole.

(a) Briefly describe the setup: What is the constraint? What gauge field acts on the particle?

(b) Why does the Aharonov-Bohm phase \(\phi = \frac{2\pi qg}{\hbar c}\) restrict the allowed eigenstates on the sphere? (Hint: consider the eigenvalue of the angular momentum operator in the \(z\)-direction.)

(c) If \(qg = \frac{\hbar c}{2}\) (minimal monopole), what is the minimum value of \(|m_z|\) for the orbital angular momentum? How does this compare to the usual quantum rotor (no monopole)?

7. Gauge Transformation and Physical Observables

The location of the Dirac string is arbitrary. Show that physical quantities are independent of where the string is placed.

(a) Suppose we rotate our coordinate system so that the string now points along a different direction (still extending to infinity from the monopole). How does the vector potential change? (You need not compute this explicitly; describe the transformation conceptually.)

(b) When a charged particle encircles the monopole, it acquires a phase. Show that this phase is independent of the string location by demonstrating that different string placements give the same Aharonov-Bohm phase for a path that avoids the string.

(c) In general, why must gauge-dependent quantities (like vector potential or string position) not appear in physically measurable predictions? Give an example from this lecture.

8. Derivation Challenge: Verify the Curl

Verify that the north-patch vector potential:

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

yields \(\nabla \times \boldsymbol{A}^{(N)} = \frac{g}{r^2}\hat{r}\) away from the string.

Hint: Use the formula for curl in spherical coordinates:

\[ \nabla \times \boldsymbol{F} = \frac{1}{r\sin\theta}\left(\frac{\partial(F_\phi \sin\theta)}{\partial\theta} - \frac{\partial F_\theta}{\partial\phi}\right)\hat{r} + \ldots\]

and note that \(\boldsymbol{A}\) has only a \(\phi\)-component.

9. Quantization Without Monopoles (Conceptual Challenge)

In the Standard Model, electric charge quantization is an axiom: we simply postulate that charges are quantized. Dirac’s monopole argument provides an alternative explanation: charge quantization emerges from topology (single-valuedness of the wavefunction).

(a) What are the conceptual advantages of Dirac’s explanation over simply postulating charge quantization?

(b) If no monopoles exist in our universe, does Dirac’s argument lose its force? (Consider whether the logical possibility of monopoles constrains the theory.)

(c) In some condensed matter systems (e.g., spin ice), monopole-like quasiparticles emerge. Do you expect the same quantization condition to apply? Why or why not?

10. Real-World Monopole Searches

Magnetic monopoles have been searched for in particle accelerators, cosmic rays, and superconducting detectors, with no confirmed detection to date.

(a) If a monopole is discovered, what does Dirac’s quantization condition tell us? Specifically, given the elementary charge \(e\), what would be the magnetic charge \(g\) of the smallest monopole?

(b) Estimate the energy scale at which monopoles might appear. (Hint: In grand unified theories, monopoles are related to the unification scale, roughly \(10^{16}\) GeV. Compare this to accessible accelerator energies today, \(\sim 10^2\)–\(10^4\) GeV.)