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The Search for Magnetic Monopoles Arttu Rajantie Physics Today The search for magnetic monopoles Arttu Rajantie Citation: Physics Today 69(10), 40 (2016); doi: 10.1063/PT.3.3328 View online: http://dx.doi.org/10.1063/PT.3.3328 View Table of Contents: http://scitation.aip.org/content/aip/magazine/physicstoday/69/10?ver=pdfcov Published by the AIP Publishing Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 189.60.119.130 On: Mon, 17 Oct 2016 21:25:56 The search for MAGNETIC MONOPOLES Arttu Rajantie The discovery of the mysterious hypothetical particles would provide a tantalizing glimpse of new laws of nature beyond the standard model. HEIKKA VALJA/MoEDAL COLLABORATION Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 189.60.119.130 On: Mon, 17 Oct 2016 21:25:56 Arttu Rajantie is a professor in the department of physics at Imperial College London and a member of the MoEDAL collaboration. lectricity and magnetism appear everywhere in the monopoles would attract each other and modern world and form the basis of most of our like monopoles would repel each other; their trajectories would bend in an elec- technology. Therefore, it would be natural to assume tric field, and so forth. Indeed, a hypo- that they are already fully understood and no longer thetical universe in which all electric pose unanswered fundamental physics questions. Indeed, charges were replaced by magnetic for most practical purposes they are perfectly well described by charges of the same strength would be completely indistinguishable from ours. Eclassical electrodynamics, as formulated by James Clerk Maxwell The word “electric” is simply a label for in 1864. At a deeper level, a consistent quantum mechanical account the type of charge that exists, and the is given by quantum electrodynamics, part of the standard model word “magnetic” for the type that ap- parently does not. Of course, if they both of particle physics. The theory works so well that it predicts existed, there would be genuinely new the magnetic dipole moment of the electron accurately to 10 electromagnetic phenomena. significant figures. Nevertheless, there is still an elementary aspect If Maxwell’s theory of electrodynam- of electromagnetism that we do not understand: the question of ics is perfectly compatible with magnetic 1 charges, why did he not include them? magnetic monopoles. Simply because experiments at the time suggested that the charges didn’t exist. That magnets always have two poles—north and south— Those experiments were simple by today’s standards—they in- seems like an obvious empirical fact. Yet we do not know any volved things like floating magnets and cutting long magnets theoretical reason why magnetic monopoles, magnets with a to pieces. But in spite of vast improvements in technology and single north or south pole, could not exist. Are we still missing experimental techniques, his assumption still seems to hold. some crucial fundamental aspect of the theory? Or do magnetic And despite equally dramatic advances in our understanding monopoles exist and we simply have not managed to find of fundamental physics, we still don’t understand why. them yet? Quantum quandary Magnetic mystery At first sight, magnetic monopoles seem to be incompatible Nothing in classical electrodynamics prohibits magnetic mono - with quantum mechanics. That is because in quantum mechan- poles; in fact, they would make the theory more symmetric. As ics, electromagnetic fields have to be described in terms of a Maxwell noted, the laws governing electricity and magnetism scalar potential ϕ and vector potential A. The magnetic field is are identical. That can be seen in the Maxwell equations of elec- given by the curl of the vector potential, B = ∇ × A, and it fol- trodynamics, which in vacuum have a duality symmetry—the lows from elementary vector calculus that the field must then electric terms can be replaced with magnetic terms, and vice be sourceless, ∇ · B = 0. In other words, magnetic field lines versa, in such a way that the equations are left unchanged. That cannot end. So how can there be magnetic monopoles? symmetry is broken only in the presence of electric charges and The same problem appears in classical electrodynamics, currents, which have no magnetic counterparts. If magnetic where potentials are often used as mathematical tools. But in monopoles existed, they would carry the magnetic equivalent classical physics the use of potentials is optional, because any of an electric charge, and they would restore the duality sym- system can be described using electric and magnetic fields in- metry (see figure 1). On aesthetic grounds, one would therefore stead. By contrast, in quantum physics the potentials couple expect their existence. directly to the complex phase of the quantum wavefunction— The duality symmetry provides clues to the likely traits of with real physical consequences—and therefore their use can- the hypothetical magnetic charges and currents. They would not be avoided. behave in exactly the same way as electrically charged parti- In 1931, however, British physicist Paul Dirac ingeniously cles: The magnetic charge would be conserved, so the lightest showed that the requirement for unbroken magnetic field magnetic monopole would be a stable particle; opposite lines doesn’t rule out monopoles.2 Quantum mechanics allows OCTOBER 2016 | PHYSICS TODAY 41 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 189.60.119.130 On: Mon, 17 Oct 2016 21:25:56 MAGNETIC MONOPOLES them to exist, but only if their mag- can use perturbation theory when- netic charge has exactly the right ever interactions are weak. For strength. electrically charged particles, the In Dirac’s model, illustrated in interaction strength is character- figure 2, each magnetic north pole ized by the fine-structure con- 2 is connected to a magnetic south stant α = e /4πħcϵ0 ≈ 1/137. Because pole by a line of singularity called α is much smaller than one, per - a Dirac string. That string is effec- turbation theory works well. By tively an idealized solenoid with contrast, the Dirac quantization zero thickness, and it carries mag- condition implies a magnetic fine- netic flux from the south pole to the north FIGURE 1. MAXWELL’S EQUATIONS structure constant that is much greater than 2 pole so that the field lines remain continu- OF ELECTRODYNAMICS, amended one: αM = μ0g /4πħc = 1/4α ≈ 34. Perturbation ous. In classical physics, such a string would to include magnetic monopoles. theory is therefore not applicable. That said, be easily observable because of the effect it The terms on the right-hand sides of the same is true for strong nuclear forces, but would have on electrically charged particles. the equations at right arise due to physicists have made progress on that front But in quantum physics, if the magnetic magnetic monopoles. The arrows nonetheless; instead of perturbation theory, charge g of the poles has exactly the right indicate transformations that obey they use numerical lattice Monte Carlo sim- value, electrically charged particles are un- duality symmetry. Here, E and B are ulations. Similar methods are being devel- affected by the string’s presence. That value the electric and magnetic fields, oped for magnetic monopoles.3 is given by the so-called Dirac quantization respectively; ϵ0 and μ0 are the permit- Another, deeper problem is that mono - condition tivity and permeability of vacuum; c poles must be attached to Dirac strings. J is the speed of light; ρ and are the Quantum fields describe point-like elemen- electric charge and current densities; tary particles, but the Dirac string is a line- and ρ and J are the magnetic M M like extended object. Including it in the the- charge and current densities. where e is the electric charge of a particle ory in a way that respects its fundamental used to probe the string, ħ is Planck’s con- symmetries—the Lorentz invariance of spe- stant, and n is any integer. If the Dirac con- cial relativity and the gauge invariance of dition is satisfied by the electric charges of all particles, no ex- electrodynamics—is difficult. Furthermore, the Dirac string periment can observe the string. Thus, Dirac argued, the string has to be included in such a way that it becomes completely is not really there: It is a mathematical artifact, a consequence unobservable if the Dirac quantization condition is satisfied. of the variables chosen for the theoretical description. Only the Although theoretical formulations dating back to the 1960s two poles at the ends of the string are real, and physically they satisfy those criteria, they are cumbersome and haven’t been appear as two separate particles—free magnetic monopoles. studied much. The quantization condition has important consequences. Could the difficulty of finding a field theoretical description First, it tells us what the charge g of a magnetic monopole of magnetic monopoles reflect some deeper fundamental in- should be, which turns out to be very strong: The magnetic compatibility with quantum field theory? Again, the answer is force between two monopoles, each with a single Dirac charge no. In 1974 Gerard ’t Hooft and Alexander Polyakov found so- gD of 2πħ/μ0e, would be 4700 times as strong as the Coulomb called hedgehog solutions for quantum field theories in which force between two electrons. the electromagnetic field is a part of a larger unified interaction.4 Second, the condition implies that if monopoles exist, the Such solutions are “lumps” of field with finite, nonzero size, electric charge must be quantized. In other words, all particles each lump consisting of a large number of elementary quanta.
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