Dirac Magnetic Monopole Quantization

Equivalent forms

\frac{e g}{2\pi\hbar} = n,\ n\in\mathbb{Z}

g_D = \frac{h}{e}\ (n=1)

\frac{g}{e} = \frac{1}{2\alpha}n

One hypothetical particle, never yet seen, would explain a deep fact we observe everywhere — that charge comes in exact integer units. Few 'if' statements in physics buy so much.

Symbol	Name	SI unit	Dimension	Range
$n$	Quantum number Integer in the quantization condition	-	1	1 – 5
$g$	Magnetic chargeoutput Monopole magnetic charge (flux it emits)	Wb	M·L²·T⁻²·I⁻¹	4.136e-15 – 2.07e-14
$e$	Elementary charge Electric charge quantum	C	I·T	1.602176634e-19 – 1.602176634e-19

Unit systems

Symbol	SI	CGS	Imperial
$n$	-	-	-
$g$	Wb	-	Wb
$e$	C	statC	C

Where it holds

A theoretical result of quantum mechanics in a monopole background; assumes standard QM single-valuedness. 't Hooft–Polyakov monopoles arise in grand unified theories with calculable mass

\sim 10^{16} GeV

; none have been observed (MoEDAL, MACRO set limits).

Dimensional analysis

$eg = nh$ : $[e][g] = C\cdot Wb$ = $(A\cdot s)(V\cdot s) = J\cdot s = [h] \checkmark$

Discovery

Paul Dirac · 1931

Dirac asked what a point magnetic charge would do to quantum mechanics. Insisting the electron wavefunction stay single-valued around the monopole's 'string' of flux, he derived eg = nℏ/2 (his convention) — a stunning explanation for the observed quantization of electric charge from the mere possible existence of one monopole.

Try this

Why is electric charge quantized — always an exact multiple of e? Dirac showed a single magnetic monopole, anywhere in the universe, would force it.

Derive the Dirac quantization condition relating electric charge e and magnetic charge g, and compute the minimal magnetic charge.

Research status: speculative

Real-world applications

Explains observed quantization of electric charge
Central to grand unified theories and early-universe cosmology
Spin-ice 'emergent monopoles' realize the math in condensed matter
Drives ultra-sensitive SQUID and LHC detector design

Common misconceptions

Monopoles have been found — only unconfirmed single events and limits exist
Maxwell's equations forbid them — they forbid them only as written; adding magnetic charge symmetrizes the equations
The condition fixes g uniquely — it fixes the product eg in integer steps, given e

Experimental verification

None confirmed. Searches include Blas Cabrera's 1982 SQUID event (a single tantalizing flux jump, never repeated), the MACRO and IceCube cosmic searches, and MoEDAL at the LHC — all setting upper limits on monopole flux and production.

Derivation

A monopole of charge g produces total magnetic flux

\Phi

= µ

_{0} g

over a sphere.

Describing its vector potential needs a singular 'Dirac string'; requiring the string to be unobservable means the Aharonov–Bohm phase

e\Phi /\hbar

around it be a multiple of

2\pi

:

e\cdot g/\hbar = 2\pi n

, i.e.

eg = 2\pi n\hbar = nh

, so

g = nh/e

.

Limiting cases

n = 1

⟶

g = h/e \approx 4.14\times 10^{-15} Wb

The minimal Dirac charge — one flux quantum h/e of outgoing magnetic flux.

No monopole exists⟶ charge need not quantizeDirac's argument is one-way: a monopole forces quantization, but quantization can also arise from grand unification.

Coupling

g/e \approx 1/2\alpha

⟶

\approx 68.5

The monopole's magnetic coupling is huge — it would ionize matter ferociously, guiding searches.

What if…

What if even one monopole exists anywhere?

Dirac's condition then forces every electric charge in the universe to be an integer multiple of e — explaining charge quantization.

What if we add magnetic charge to Maxwell's equations?

They become symmetric under E↔B: $\nabla \cdot B$ = µ $_{0}\rho _m$ $and \nabla \times E$ gains a magnetic-current term — mathematically elegant duality.

What if a monopole passed through a superconducting loop?

It would induce a permanent, quantized two-flux-quantum step in the loop current — exactly the Cabrera SQUID signature being hunted.

1

Minimal magnetic charge

Given ·

n:: 1
h:: 6.62607015e-34
e:: 1.602176634e-19

Find · g

Steps

$Set n = 1$ in $g = nh/e$
$g = 6.62607\times 10^{-34} / 1.60218\times 10^{-19}$
$g \approx 4.14\times 10^{-15} Wb$ — exactly one magnetic flux quantum h/e

Result ·

g = nh/e = 6.626\times 10^{-34} / 1.602\times 10^{-19} \approx 4.14\times 10^{-15} Wb

Maxwell's Equations (Unified)→Gauss's Law for Magnetism→Faraday's Law of Induction→