Editing Magnetic monopole (section)

=== Dirac string ===
{{Main|Dirac string}}
A [[gauge theory]] like electromagnetism is defined by a gauge field, which associates a group element to each path in space time. For infinitesimal paths, the group element is close to the identity, while for longer paths the group element is the successive product of the infinitesimal group elements along the way.

In electrodynamics, the group is [[U(1)]], unit complex numbers under multiplication. For infinitesimal paths, the group element is {{math|1 + ''iA''<sub>''μ''</sub>''dx''<sup>''μ''</sup>}} which implies that for finite paths parametrized by {{math|''s''}}, the group element is:
{{block indent|1=<math>\prod_s \left( 1+ieA_\mu {dx^\mu \over ds} \, ds \right) = \exp \left( ie\int A\cdot dx \right) . </math>}}

The map from paths to group elements is called the [[Wilson loop]] or the [[holonomy]], and for a U(1) gauge group it is the phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

{{block indent|1=<math>e \oint_{\partial D} A\cdot dx  = e \int_D (\nabla \times A) \, dS = e \int_D B \, dS.</math>}}

So that the phase a charged particle gets when going in a loop is the [[magnetic flux]] through the loop. When a small [[solenoid]] has a magnetic flux, there are [[Aharonov–Bohm effect|interference fringes]] for charged particles which go around the solenoid, or around different sides of the solenoid, which reveal its presence.

But if all particle charges are integer multiples of {{math|''e''}}, solenoids with a flux of {{math|2{{pi}}/''e''}} have no interference fringes, because the phase factor for any charged particle is {{math|1=exp(2{{pi}}''i'') = 1}}. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of {{math|2{{pi}}/''e''}}, when the flux leaked out from one of its ends it would be indistinguishable from a monopole.

Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of the solenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles of opposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable, so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular by sliding the string to where it cannot be seen.