Magnetic monopoles

A magnetic monopole is a particle that carries magnetic charge alone, a north pole without a south pole. Dirac showed in 1931 that quantum mechanics allows such a particle only if the product of its magnetic charge with every electric charge is quantized, and gauge theories whose symmetry breaks in a suitable pattern contain monopoles as stable, heavy configurations of their fields. The early universe should have made many of them, yet every search has come up empty, and the standard account of that absence is inflation. This section holds that established physics; the program’s own monopole claims are in the parent section, and its retired monopole dark-matter candidate is filed with the dark-matter candidates.

Literature only: all six pages are established physics [T], so the section ranks no claims of the program’s own; those are tracked in the registers of claims that stand, the graveyard, open problems and the queue, conjectures and superseded statements. Each page ends with a section on how the program used the result, mostly in #14, the withdrawn soliton manuscript, and links the program page that carries the use. The number from this section that the current gate checks, the Kibble normalization p ≈ 104 [V], belongs to Obituary: the Kibble normalization and is gated by the soliton-obituary script.

Map#

The section runs from the quantum condition on a single monopole, through the monopoles of a broken gauge symmetry, the charge a vacuum angle gives them and their production in the early universe, to the problem that production creates and the observations that bound it.

Dirac quantization and the DSZ pairing comes first because every later page uses it. It derives Dirac’s condition e·g ∈ 2πℤ three ways (the invisibility of the Dirac string, the two-patch construction, the angular momentum of the field), explains why the literature writes it with 2π or with 4π and what that does to a coupling, and extends it to dyons as the Dirac–Schwinger–Zwanziger pairing ⟨q, q′⟩ = e₁g₂ − e₂g₁, an antisymmetric integer form whose zero set defines mutual locality. It ends with the consequence that a weakly coupled theory has strongly coupled monopoles, the result that survived the soliton candidate.

The ’t Hooft–Polyakov monopole is the concrete monopole of a gauge theory whose SU(2) symmetry breaks to U(1): stable because its Higgs field winds once around the sphere of vacua (π₂(S²) = ℤ), absent from the electroweak theory (π₂(S³) = 0), with magnetic charge 4π/g_d and mass M = 4πf v/g_d. The page follows the mass from the Bogomolny bound, f = 1, to the value f ≈ 1.79 at large scalar self-coupling, with the round-3 referee’s own numerical solution.

The Witten effect is the electric charge such a monopole carries when the gauge theory has a nonzero vacuum angle θ: θ/2π of the unit of charge for each unit of magnetic charge, which makes the monopole a dyon. The pairing of the dyon lattice does not depend on θ, and raising θ by 2π relabels the dyons, the shift T of SL(2, ℤ). The program met the effect once: ruling 9 of round 3 accepted, without re-checking it, that the retired soliton candidate’s hidden monopoles would carry a small visible charge if the sectors mixed, and strict sector separation closed that channel.

The Kibble mechanism counts the defects a cooling universe produces at such a transition: the homotopy classification of walls, strings and monopoles, the causal bound of about one defect per horizon volume and its conversion to the yield n/s = p (T_c/M_Pl)³, and the two mechanisms that only raise the count, Kibble–Zurek freeze-out and bubble nucleation. The monopole problem is the consequence for superheavy monopoles: their density grows as the fourth power of the breaking scale, annihilation cannot remove enough of them even at strong coupling, and inflation is the standard remedy; the same physics in a hidden sector rules out monopole dark matter from breaking scales above about 100 PeV unless the monopoles are diluted. The monopole flux bounds close the section with the observational limits: the Parker bound from the survival of the Galaxy’s magnetic field, the extended bound from its seed field, and the direct search of the Monopole, Astrophysics and Cosmic Ray Observatory.

Boundaries#