Dirac Monopole Preview

A Dirac monopole is a hypothetical magnetic charge whose magnetic field is radial (outward for $g>0$):

$$\mathbf B(\mathbf r) = g\frac{\mathbf r}{r^3}, \qquad r>0.$$

The total flux through a sphere surrounding the origin is

$$\int_{S^2} \mathbf B\cdot d\mathbf S = 4\pi g.$$

The quantum question is how to describe a charged wavefunction consistently in this background. The wavefunction is specified in overlapping patches, with phase relations between them. Consistency quantizes the product of electric and magnetic charges. This is a conditional statement about a hypothetical monopole, not a proof that monopoles exist. The same geometry appears in Berry curvature and Chern numbers.

Why a Vector Potential Is Subtle

Away from the origin,

$$\nabla\cdot\mathbf B=0,$$

but the total flux through a sphere enclosing the origin is nonzero. A globally smooth vector potential on the punctured space would imply, by Stokes’ theorem, that the flux through a closed sphere vanishes:

$$\int_{S^2} \mathbf B\cdot d\mathbf S = \int_{S^2} dA = \int_{\partial S^2}A = 0.$$

The last step uses $\partial S^2=\varnothing$. Therefore a nonzero monopole flux cannot be described by one smooth globally defined vector potential on the sphere surrounding the origin.

The resolution is not to abandon vector potentials. It is to use more than one gauge patch.

Two-Patch Potentials

On a sphere of fixed radius, use polar coordinates $(\theta,\phi)$. A standard pair of vector-potential one-forms is

$$A_N = g(1-\cos\theta)\,d\phi,$$

regular away from the south pole, and

$$A_S = -g(1+\cos\theta)\,d\phi,$$

regular away from the north pole. These are one-forms: $A=\mathbf A\cdot d\mathbf r$, not the azimuthal vector component itself. For example,

$$\mathbf A_N=\frac{g(1-\cos\theta)}{r\sin\theta}\,\hat{\boldsymbol\phi}.$$

The factor $r\sin\theta$ in the line element converts this vector expression into $A_N$. Both patches give the same local magnetic field:

$$dA_N = dA_S = g\sin\theta\,d\theta\wedge d\phi.$$

On the overlap of the northern and southern patches,

$$A_N-A_S = 2g\,d\phi = d(2g\phi).$$

Thus the two potentials differ by a gauge transformation with gauge function

$$\chi = 2g\phi.$$

The price is that $\phi$ is not a globally single-valued coordinate. That is exactly where quantization enters.

Dirac Quantization

For a particle of electric charge $q$, the wavefunction transforms under

$$A\mapsto A+d\chi$$

as

$$\psi \mapsto \exp\left( \frac{iq\chi}{\hbar} \right)\psi$$

with the minimal-coupling convention $p\mapsto p-qA$. On the overlap of the two monopole patches,

$$\psi_N = \exp\left( \frac{iq(2g\phi)}{\hbar} \right) \psi_S.$$

As $\phi$ goes from $0$ to $2\pi$, the transition phase changes by

$$\exp\left( \frac{iq\,4\pi g}{\hbar} \right).$$

For the wavefunction patching to be single-valued, this factor must be $1$:

$$\exp\left( \frac{i4\pi qg}{\hbar} \right) = 1.$$

Therefore

$$\frac{4\pi qg}{\hbar} = 2\pi n, \qquad n\in\mathbb Z,$$

or

$$2qg = n\hbar.$$

This is the Dirac quantization condition in the convention $\mathbf B=g\mathbf r/r^3$. Unit systems and definitions of $g$ differ across books, so the invariant statement is that the phase around the patch overlap must be single-valued:

$$\frac{q}{2\pi\hbar} \int_{S^2} \mathbf B\cdot d\mathbf S \in \mathbb Z.$$

Dirac String View

The two-patch description is the clean global picture. A one-patch description can also be used, but it contains a line singularity called a Dirac string. For example, $A_N$ is regular at the north pole but singular along the negative $z$ axis.

The string is not meant to be a physical solenoid in the ideal monopole theory. It is a gauge artifact if the quantization condition holds. The extra phase attributable to moving the unphysical string is set by its full flux $4\pi g$. Quantization makes this contribution trivial:

$$\exp\left( \frac{iq\,4\pi g}{\hbar} \right) = 1.$$

This does not make every finite loop phase equal to one: a finite loop can enclose physical monopole flux. It makes the two patch descriptions agree, so interference cannot detect the placement of the string. If the condition failed, the string would become physically observable, which would contradict the idea that it is only a gauge singularity.

Relation to Aharonov–Bohm Phase

The Aharonov–Bohm effect teaches that the phase

$$\exp\left( \frac{iq}{\hbar} \oint_C \mathbf A\cdot d\mathbf r \right)$$

is physical around a closed loop. The Dirac monopole applies the same lesson to a loop encircling the would-be string or, more invariantly, to a patch overlap around the equator.

Dirac quantization makes the difference between the two patch integrals an integer multiple of $2\pi$ in phase units. Their common loop phase can still be nontrivial, as the equator exercise below illustrates.

This is a useful shift in perspective:

• Aharonov–Bohm effect: a nontrivial loop can detect enclosed flux.
• Dirac monopole: consistency requires the patching phase around the equator to be single-valued.
• Chern number: the normalized flux through a closed surface is an integer.

Bundle Interpretation

The monopole field is the curvature of a $U(1)$ connection on a nontrivial line bundle over $S^2$. The normalized flux is the first Chern number:

$$c_1 = \frac{q}{2\pi\hbar} \int_{S^2} \mathbf B\cdot d\mathbf S.$$

Using the monopole flux,

$$c_1 = \frac{q}{2\pi\hbar} 4\pi g = \frac{2qg}{\hbar}.$$

Dirac quantization is the statement

$$c_1\in\mathbb Z.$$

This is why monopoles are not merely exotic sources. They are the simplest physical doorway into the idea that gauge fields can live on topologically nontrivial bundles.

Berry Monopole Analogy

The spin-$1/2$ Berry phase has an effective monopole in parameter space. For the eigenstate of $\boldsymbol\sigma\cdot\mathbf n$ with eigenvalue $+1$, represented by $(\cos(\theta/2),e^{i\phi}\sin(\theta/2))^{\mathsf T}$, and connection $\mathcal A=i\langle +|d+\rangle$,

$$F_+ = -\frac12 \sin\theta\,d\theta\wedge d\phi.$$

The flux through the parameter sphere is

$$\int_{S^2}F_+ = -2\pi,$$

so the Berry Chern number is

$$C_+ = \frac{1}{2\pi} \int_{S^2}F_+ = -1.$$

This is called a Berry monopole because the degeneracy at the origin of parameter space acts like a source of curvature flux. It is not an electromagnetic monopole in ordinary space. The analogy is geometric: both cases involve a connection, curvature, patching, and quantized flux through a closed sphere.

Common Mistakes

• Treating the Dirac string as a physical object after imposing the quantization condition.
• Expecting one globally smooth vector potential for a field with nonzero flux through a closed sphere.
• Forgetting that the numerical form of $qg$ quantization depends on unit conventions.
• Confusing the Berry monopole in parameter space with an electromagnetic monopole in real space.
• Calling any radial-looking Berry curvature a monopole without checking the flux through a closed surface.
• Applying Stokes’ theorem with a single gauge potential on a surface that cannot be covered by one smooth patch.
• Thinking the Dirac argument proves monopoles exist; it proves a consistency condition if they do.

Exercises

1. Compute the monopole flux.

For

$$\mathbf B = g\frac{\mathbf r}{r^3},$$

show that the flux through a sphere of radius $R$ is $4\pi g$.

Solution

On the sphere,

$$\mathbf B = \frac{g}{R^2}\hat{\mathbf r}, \qquad d\mathbf S = R^2\sin\theta\,d\theta\,d\phi\,\hat{\mathbf r}.$$

Therefore

$$\begin{aligned} \int_{S^2}\mathbf B\cdot d\mathbf S &= \int_0^{2\pi} \int_0^\pi g\sin\theta\,d\theta\,d\phi\\ &= 4\pi g. \end{aligned}$$

2. Derive the quantization condition from patching.

Use $A_N-A_S=d(2g\phi)$ and $\psi_N=e^{iq(2g\phi)/\hbar}\psi_S$ to show that $2qg=n\hbar$.

Solution

When $\phi$ increases by $2\pi$, the transition phase changes by

$$\exp\left[ \frac{iq}{\hbar} 2g(2\pi) \right] = \exp\left( \frac{i4\pi qg}{\hbar} \right).$$

Single-valued patching requires this to be $1$, so

$$\frac{4\pi qg}{\hbar} = 2\pi n, \qquad n\in\mathbb Z.$$

Thus

$$2qg=n\hbar.$$

3. Evaluate the Wilson phase around the equator using each potential, assuming $2qg=n\hbar$. Must it equal one?

Solution

At $\theta=\pi/2$, the integrals with increasing $\phi$ are

$$\oint A_N=2\pi g,\qquad \oint A_S=-2\pi g.$$

Therefore the two phases are $e^{i\pi n}$ and $e^{-i\pi n}$, both equal to $(-1)^n$. Their difference in the exponent is $2\pi n$, so the patches agree. For odd $n$, the physical equatorial holonomy is $-1$, not $1$. Consistent patching removes the artificial string contribution without removing the real magnetic field.

4. Compare to the spin-$1/2$ Berry monopole.

For

$$F_+ = -\frac12\sin\theta\,d\theta\wedge d\phi,$$

compute the Chern number over the sphere.

Solution

The curvature integral is

$$\int_{S^2}F_+ = \int_0^{2\pi} \int_0^\pi -\frac12\sin\theta\,d\theta\,d\phi = -2\pi.$$

Therefore

$$C_+ = \frac{1}{2\pi} \int_{S^2}F_+ = -1.$$

With the displayed eigenstate, connection, and outward sphere orientation, the sign is fixed. Choosing the opposite spin eigenstate or reversing the surface orientation reverses it. A smooth phase change of the same eigenstate cannot change its curvature or Chern number.

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.
• P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proceedings of the Royal Society A 133, 60–72, 1931.
• M. Nakahara, Geometry, Topology and Physics, 2nd ed., CRC Press, 2003.
• J. Preskill, “Magnetic monopoles,” Annual Review of Nuclear and Particle Science 34, 461–530, 1984.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167–2170, 1983.
• T. T. Wu and C. N. Yang, “Concept of nonintegrable phase factors and global formulation of gauge fields,” Physical Review D 12, 3845–3857, 1975.