In 1931 Dirac showed that quantum mechanics is consistent with isolated magnetic poles provided the pole strength is quantized, eμ0=ℏc/2e\mu_0=\hbar c/2, and that the existence of even one such pole would explain why all electric charges are integer multiples of the elementary charge ee. Dirac himself emphasized that the theory fails to give the 'purely electronic' condition he sought for the value ℏc/e2≈137\hbar c/e^2\approx 137, and that non-observation of poles was accounted for only speculatively (the 4692144692\tfrac{1}{4}-fold attraction between opposite poles). Ninety-five years of searches (matter samples, cosmic rays, colliders, and heavy-ion collisions exploiting the dual Schwinger mechanism of quantum tunnelling) and an extensive theory literature ('t Hooft-Polyakov monopoles predicted by grand unified theories, electroweak Cho-Maison monopoles, duality) have produced no confirmed monopole and no accepted derivation of the fine-structure constant. The problem remains open: determine whether isolated magnetic monopoles exist in nature with charges obeying the Dirac quantization condition, and thereby whether the observed quantization of electric charge receives its explanation from Dirac's argument, together with the distinct but related objective Dirac attached to it — whether the numerical value of the elementary charge, ℏc/e2≈137\hbar c/e^2\approx 137, can be derived from a theory in which ee is not an input.

Background

Classical electrodynamics is built on the assumption that magnetic field lines never end: the equation ∇⋅B=0\nabla\cdot\mathbf{B}=0 forbids isolated magnetic charges (magnetic monopoles), while electric charges exist and act as sources of E\mathbf{E}. In quantum mechanics the electromagnetic field enters through the vector potential A\mathbf{A} with B=∇×A\mathbf{B}=\nabla\times\mathbf{A}, which at first sight also excludes monopoles.

In 1931 Dirac reconsidered this within his framework of wave functions with non-integrable phase (a phase that depends on the path, not just the endpoints). Wave functions may vanish along nodal lines (curves on which the wave function is identically zero), and the endpoints of such lines act as magnetic poles. Requiring the phase to be single-valued for a charged particle moving in the field of a pole of strength μ0\mu_0 yields the Dirac quantization condition eμ0=nℏc/2e\mu_0=n\hbar c/2 for integer nn, so the minimal pole strength is μ0=ℏc/(2e)≈68.5e\mu_0=\hbar c/(2e)\approx 68.5e (using ℏc/e2≈137\hbar c/e^2\approx 137). Equivalently, every electric charge must be an integer multiple of ee: the existence of a single magnetic pole would explain the observed quantization of electric charge, one of the most striking unexplained regularities of nature. The singular line (the 'Dirac string') that carries flux to the pole is unobservable exactly when the quantization condition holds. Dirac noted that the force between two one-quantum poles of opposite sign is 4692144692\tfrac{1}{4} times that between electron and proton and speculated that this 'may perhaps account for why poles of opposite sign have never yet been separated'; and he found it 'rather disappointing' that the theory yields this electricity-magnetism reciprocity rather than 'a purely electronic quantum condition such as ℏc/e2\hbar c/e^2' — the value of the fine-structure constant α−1=ℏc/e2≈137\alpha^{-1}=\hbar c/e^2\approx 137, which sets the strength of the electromagnetic interaction, remained unexplained.

Later theory transformed but did not settle the question. In 1974 't Hooft and Polyakov independently showed that any grand unified theory (GUT; a gauge theory unifying the strong and electroweak interactions, whose symmetry breaks down to leave electromagnetism) necessarily contains topological soliton monopoles (stable, localized, particle-like field configurations) satisfying the Dirac condition, with masses of order 101710^{17} GeV, far beyond any collider; the electroweak Cho-Maison monopole (the analogue solution in the electroweak sector of the Standard Model) may be as light as a few TeV. Polchinski discussed the converse of Dirac's implication, proposing 'principles of completeness' of the particle spectrum, under which any consistent framework in which electric charge is quantized will contain magnetic monopoles. Electric-magnetic duality symmetries (symmetries relating the electric and magnetic degrees of freedom of a theory) make monopoles central to modern field theory and string theory. On the experimental side, monopoles have been sought in ocean sediments, moon rocks and iron ores, in cosmic rays (the MACRO, IceCube and ANTARES detectors), in proton-proton collisions at the Large Hadron Collider (LHC, the highest-energy particle accelerator, at CERN) with the ATLAS and MoEDAL detectors, and via the dual Schwinger mechanism (pair production by quantum tunnelling in the ultra-strong magnetic fields of heavy-ion collisions and of magnetars (neutron stars carrying the strongest magnetic fields known)); no confirmed detection exists. In spin-ice frustrated magnets (crystals whose magnetic moments cannot all satisfy their pairwise interactions simultaneously), emergent quasiparticles carrying effective magnetic charge have been observed and reported to obey a Dirac-Schwinger quantization pattern, but these are excitations inside a medium, not isolated poles in vacuum. The fine-structure constant is now measured to 81 parts per trillion, α−1=137.035999206(11)\alpha^{-1}=137.035999206(11), yet remains an input parameter of the Standard Model with no accepted theoretical derivation.

Problem Statement

Determine whether isolated magnetic monopoles exist in nature as physical particles — elementary or solitonic — carrying magnetic charges that are integer multiples of the minimal Dirac charge μ0=ℏc/2e\mu_0=\hbar c/2e (equivalently eμ=nℏc/2e\mu=n\hbar c/2; in units ℏ=c=1\hbar=c=1, gD=2π/e≈20.7g_D=2\pi/e\approx 20.7), and thereby whether the observed quantisation of electric charge (free charges in integer multiples of ee, with confined quarks carrying multiples of e/3e/3) receives its explanation from Dirac's argument. The problem has three linked components descending from Dirac's 1931 paper: (1) existence and detectability — determine whether isolated magnetic monopoles are observed (directly or through production), or whether there is a principled reason why none exists or is observable; (2) explanation of charge quantisation — if monopoles exist, verify that the Dirac quantization condition accounts for the observed charge lattice; if they do not, identify what does; (3) the numerical value — determine whether, and by what route, the value that makes ℏc/e2=α−1≈137.036\hbar c/e^2=\alpha^{-1}\approx 137.036 (Dirac's sought 'purely electronic quantum condition'), consistent with the measured 137.035999206(11)137.035999206(11), can be derived from a theory in which the elementary charge is not an input. These components are jointly bound by Dirac's original objective — the monopole was proposed as the mechanism that makes charge quantised, and explaining the value of ee was his motivating goal — rather than by logical implication between them, since the monopole theory itself yields the electricity-magnetism reciprocity and not a purely electronic condition for ℏc/e2\hbar c/e^2.

References

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