Dirac quantization condition

The Dirac quantization condition is the 1931 result by Paul Dirac that the existence of even a single magnetic monopole anywhere in the universe forces electric charge to be quantised. Specifically, if a magnetic monopole carries magnetic charge g and an electric particle carries charge q, the consistency of the quantum-mechanical wavefunction in the presence of both requires q g = 2 π n ℏ for some integer n. Equivalently, q g / (2 π ℏ) ∈ ℤ. The smallest nontrivial monopole charge is therefore g_D = 2 π ℏ / e ≈ 4.14 × 10^{−15} Wb, which is exactly the magnetic flux quantum Φ_0 = h / e.

Dirac's argument proceeds by considering the wavefunction of an electric charge in the field of a magnetic monopole. The vector potential A of an isolated monopole cannot be defined globally — there is necessarily a "Dirac string" emerging from the monopole, a singular line along which A blows up. The string's location is gauge-dependent; physically, two different choices of string location must give equivalent physics, which requires the integrated phase ∮ A·dℓ around any small loop encircling the string to be a multiple of 2π. This forces e g = 2 π n ℏ. The deep consequence is that the empirical fact of electric-charge quantisation — the otherwise-mysterious observation that every elementary particle has charge that is an integer multiple of e/3 — becomes a theoretical consequence of monopole existence. The flip side is that the absence of observed monopoles after sixty years of dedicated search (MoEDAL at the LHC, lunar-regolith analyses, dedicated solenoid experiments) is one of the puzzles motivating cosmological models with monopole inflation or with monopole production cut off below the Hubble horizon.