Dirac's magnetic monopole requires a string — a semi-infinite line of singularity extending from the monopole to infinity. The string is gauge-dependent: its position can be moved by a gauge transformation, and physical observables don't depend on where it is. This gauge invariance is what makes the monopole consistent despite the string's apparent physicality.
But the string has a self-force. Modeled as a semi-infinite solenoid with fixed magnetic flux Φ and radius a, the string generates a radial magnetic field at its surface. The current loops that constitute the solenoid interact with this field, producing a net force along the string's axis. The explicit result: F = Φ²/(2πμ₀a²).
The force diverges as a → 0. A true Dirac string — zero radius, finite flux — has infinite self-force. This is not a computational artifact but a physical consequence of concentrating magnetic flux into a line: the magnetic pressure scales as the inverse square of the radius, and the pressure has no mechanism for cancellation.
The divergence is the content. A Dirac string is unphysical not because it's a gauge artifact (it is) but because even as a physical object — a thin solenoid — it requires infinite force to hold together at zero radius. The gauge freedom that allows you to move the string is not just mathematical convenience; it's physically necessary because a physical string at any fixed position would be dynamically unstable. The string must be unphysical because the physical version destroys itself. Gauge invariance is not just permitted — it's demanded by the self-force.