Magnetic monopoles on The Broken-Born program

Dirac quantization and the DSZ pairing

Mon, 01 Jan 0001 00:00:00 +0000

Dirac quantization is the rule, found by Paul Dirac in 1931, that a magnetic monopole can exist in quantum mechanics only if the product of its magnetic charge g with every electric charge e is a whole-number multiple of a fixed constant: e·g ∈ 2πℤ in the normalization the checkpoint uses. Without the rule, the line along which the monopole’s field enters its vector potential would shift the phase of a charge carried around it, and that line would be observable. Schwinger and Zwanziger extended the rule to dyons, particles that carry both kinds of charge, as the Dirac–Schwinger–Zwanziger (DSZ) pairing of two charges, an antisymmetric integer form. Because the product of the two charges is fixed, a small electric charge forces a large magnetic one, so a weakly coupled force has strongly coupled monopoles. The program builds its charge-lattice argument on the pairing, and the weak-to-strong consequence is one of the results that survived the retired soliton candidate.

The 't Hooft–Polyakov monopole

Mon, 01 Jan 0001 00:00:00 +0000

The ’t Hooft–Polyakov monopole is a stable, particle-like configuration of fields in a gauge theory whose SU(2) symmetry is broken to U(1) by a Higgs field in the adjoint representation; far from its core it looks like a magnetic monopole of the surviving U(1). It cannot decay, because its Higgs field winds once around the space of possible vacua, a sphere, and no smooth change of the fields can undo the winding. Its mass is the symmetry-breaking scale v times 4π/g_d, with g_d the gauge coupling, times a factor f between 1 and about 1.79 set by the strength of the Higgs field’s self-interaction relative to the gauge coupling. In the program it was the object of the withdrawn manuscript #14, which proposed superheavy monopoles of this kind as dark matter; that proposal is retired, and the physics on this page is textbook.

The Witten effect

Mon, 01 Jan 0001 00:00:00 +0000

The Witten effect is the electric charge that a magnetic monopole carries in a gauge theory with a nonzero vacuum angle θ, the coefficient of a term in the Lagrangian proportional to the product E·B of the electric and magnetic fields. In such a theory a monopole’s electric charge is not a whole multiple of the unit of charge e: it is shifted by θ/2π of e for each unit of magnetic charge, so the monopole is a dyon, a particle carrying both kinds of charge. Edward Witten derived the result in 1979. Increasing θ by 2π gives back the same set of charges with the dyons relabelled, which is the shift T of the duality group SL(2, ℤ). The effect entered the program once, through the retired soliton candidate. The round-3 review accepted, without re-checking it, that a hidden vacuum angle together with kinetic mixing between the hidden photon and ours would give the candidate’s hidden monopoles a small electric charge under our photon, contradicting the manuscript’s statement that the decisive constraints act on its hidden fermion, not on the dark matter itself. Strict sector separation, the program’s reading of postulate P4, closes that channel.

The Kibble mechanism

Mon, 01 Jan 0001 00:00:00 +0000

The Kibble mechanism, proposed by Tom Kibble in 1976, is the standard account of how topological defects such as monopoles, strings and domain walls form when the early universe cools through a symmetry-breaking transition. Regions farther apart than the horizon, the distance light has travelled since the Big Bang, have had no contact, so the field settles into unrelated vacuum states in different regions, and defects are left where those regions meet. This gives at least about one defect per horizon volume, and that count is a minimum: slow transitions (Kibble–Zurek dynamics) and first-order transitions, which proceed by bubbles, make more. In the program the count decided the fate of the soliton dark-matter candidate, whose manuscript had normalized it too low and treated it as an upper end; that error and its correction are recorded with the candidate.

The monopole problem

Mon, 01 Jan 0001 00:00:00 +0000

The monopole problem, in the form the program cites (John Preskill, 1979), is the finding that the hot early universe should have produced superheavy magnetic monopoles in unacceptably large numbers, while none has ever been detected. Every grand unified theory contains such monopoles, the cooling universe makes at least about one per horizon volume when the unified symmetry breaks, and monopoles and antimonopoles annihilate one another too slowly to remove enough of them. The standard resolution is inflation, a period of rapid expansion after the monopoles form that dilutes them to an unobservable density. The program meets the problem twice: inflation is one of the two reasons its founding claim that conjugate monopoles explain the null result is dead, and round 3 ruled that the strong-coupling annihilation question of its retired soliton candidate is Preskill’s calculation, already done for hidden sectors.

Monopole flux bounds: Parker and MACRO

Mon, 01 Jan 0001 00:00:00 +0000

The monopole flux bounds are upper limits on how many magnetic monopoles pass through the Galaxy, stated as a flux per square centimetre per second per steradian. The Parker bound comes from the Galaxy’s magnetic field: monopoles moving through it would gain energy from the field and drain it faster than the galactic dynamo regenerates it, so the field’s survival caps the flux; the extended Parker bound applies the same argument to the weaker seed field of the young Galaxy. Direct searches, among them the Monopole, Astrophysics and Cosmic Ray Observatory (MACRO) whose limit the withdrawn #14 quotes, have found none. The program used these bounds only counterfactually, to show that the retired soliton candidate could not have been magnetically charged under our electromagnetism.