Non-fiction: Quantised Singularities in the Electromagnetic Field

Overview
Paul Dirac proposed a striking idea linking the existence of isolated magnetic charges to a fundamental quantum property of electric charge. He modeled a magnetic monopole as a "quantised singularity" in the electromagnetic field, a point where magnetic flux emerges and around which the usual vector potential description must be modified. By examining the quantum-mechanical behavior of an electrically charged particle in the presence of such a singularity, Dirac extracted a condition that ties the two types of charge together and suggests a deep reason why electric charge appears in discrete units.
The argument combines classical electromagnetic potentials with the quantum phase of a charged particle's wavefunction. Dirac allowed the electromagnetic vector potential to possess a line-like singularity (later called the "Dirac string") that carries the flux from the monopole to infinity. The string is a gauge artifact rather than a physical filament only if its effects on charged quantum particles are unobservable; imposing that requirement yields a quantization rule.

Key construction
The electromagnetic field of a monopole can be described everywhere except along a semi-infinite line where the vector potential becomes singular. That singular line supplies the magnetic flux that would otherwise emanate from the monopole. Classically the string is arbitrary and could be moved by a gauge transformation, but quantum mechanics makes the phase of a charged particle's wavefunction physically meaningful when it winds around the string.
Dirac focused on the phase change acquired by a charged particle transported around the string. For the string to have no observable effect, that phase change must be an integer multiple of 2π so that the wavefunction returns to itself. This single-valuedness requirement is the bridge from electromagnetic topology to quantization: it turns a gauge freedom into a quantization condition on the allowed values of charge.

Dirac quantization condition
From the phase argument Dirac derived that the product of electric charge q and magnetic charge g must be quantized. In convenient units the relation reads q g = n ħ/2, where ħ is the reduced Planck constant and n is an integer. Put differently, the existence of even a single magnetic monopole anywhere in the universe would force all electric charges to be integral multiples of a basic unit.
The condition is robust: it relies only on the wavefunction's phase behavior and the topology of space minus the string, not on detailed dynamics. The singular string can be rendered physically irrelevant only if the quantization holds, making charge quantization a topological consequence once monopoles are admitted.

Consequences and significance
Dirac's idea offered the first theoretical explanation for why electric charge is quantized, converting a puzzling empirical fact into a constraint from quantum mechanics and gauge structure. Although no elementary magnetic monopole has been observed experimentally, the conceptual payoff was large: it highlighted the physical significance of potentials and phases, and showed that global, topological properties of fields can enforce discrete quantum numbers.
The paper also planted seeds for later developments in theoretical physics. Monopoles reappear as smooth, finite-energy solutions in nonabelian gauge theories, and the Dirac quantization condition finds a natural home in modern topological and fiber-bundle descriptions of gauge fields. Condensed-matter analogs and grand-unified-theory predictions of monopoles further echo Dirac's original insight.

Legacy
Dirac's quantised singularities transformed a technical construction into a foundational principle linking topology, gauge freedom, and quantization. The Dirac monopole and the quantization condition remain central examples in teaching and research, illustrating how subtle consistency requirements in quantum mechanics can have far-reaching implications for what kinds of charges and fields are possible.