Magnetic monopole
In
Magnetism in
Some
Historical background
Early science and classical physics
Many early scientists attributed the magnetism of lodestones to two different "magnetic fluids" ("effluvia"), a north-pole fluid at one end and a south-pole fluid at the other, which attracted and repelled each other in analogy to positive and negative electric charge.[7][8] However, an improved understanding of electromagnetism in the nineteenth century showed that the magnetism of lodestones was properly explained not by magnetic monopole fluids, but rather by a combination of electric currents, the electron magnetic moment, and the magnetic moments of other particles. Gauss's law for magnetism, one of Maxwell's equations, is the mathematical statement that magnetic monopoles do not exist. Nevertheless, Pierre Curie pointed out in 1894[9] that magnetic monopoles could conceivably exist, despite not having been seen so far.
Quantum mechanics
The
Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975
Some condensed matter systems propose a structure superficially similar to a magnetic monopole, known as a flux tube. The ends of a flux tube form a magnetic dipole, but since they move independently, they can be treated for many purposes as independent magnetic monopole quasiparticles. Since 2009, numerous news reports from the popular media[16][17] have incorrectly described these systems as the long-awaited discovery of the magnetic monopoles, but the two phenomena are only superficially related to one another.[18][19] These condensed-matter systems remain an area of active research. (See § "Monopoles" in condensed-matter systems below.)
Poles and magnetism in ordinary matter
All matter isolated to date, including every atom on the periodic table and every particle in the Standard Model, has zero magnetic monopole charge. Therefore, the ordinary phenomena of magnetism and magnets do not derive from magnetic monopoles.
Instead, magnetism in ordinary matter is due to two sources. First,
Mathematically, the magnetic field of an object is often described in terms of a multipole expansion. This is an expression of the field as the sum of component fields with specific mathematical forms. The first term in the expansion is called the monopole term, the second is called dipole, then quadrupole, then octupole, and so on. Any of these terms can be present in the multipole expansion of an electric field, for example. However, in the multipole expansion of a magnetic field, the "monopole" term is always exactly zero (for ordinary matter). A magnetic monopole, if it exists, would have the defining property of producing a magnetic field whose monopole term is non-zero.
A
Maxwell's equations
Maxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the distribution of electric charge and current. The standard equations provide for electric charge, but they posit zero magnetic charge and current. Except for this constraint, the equations are symmetric under the interchange of the electric and magnetic fields. Maxwell's equations are symmetric when the charge and electric current density are zero everywhere, as in vacuum.
Maxwell's equations can also be written in a fully symmetric form if one allows for "magnetic charge" analogous to electric charge.[20] With the inclusion of a variable for the density of magnetic charge, say ρm, there is also a "magnetic current density" variable in the equations, jm.
If magnetic charge does not exist – or if it exists but is absent in a region of space – then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ∇ ⋅ B = 0 (where ∇⋅ is the
In Gaussian cgs units
The extended Maxwell's equations are as follows, in CGS-Gaussian units:[23]
Name | Without magnetic monopoles | With magnetic monopoles |
---|---|---|
Gauss's law | ||
Ampère's law (with Maxwell's extension)
|
||
Gauss's law for magnetism | ||
Faraday's law of induction | ||
Lorentz force law[23][24] |
In these equations ρm is the magnetic charge density, jm is the magnetic current density, and qm is the magnetic charge of a test particle, all defined analogously to the related quantities of electric charge and current; v is the particle's velocity and c is the speed of light. For all other definitions and details, see Maxwell's equations. For the equations in nondimensionalized form, remove the factors of c.
In SI units
In the
Maxwell's equations then take the following forms (using the same notation above):[notes 1]
Name | Without magnetic monopoles |
With magnetic monopoles | |
---|---|---|---|
Weber convention | Ampere-meter convention | ||
Gauss's law | |||
Ampère's law (with Maxwell's extension) | |||
Gauss's law for magnetism | |||
Faraday's law of induction | |||
Lorentz force equation |
Potential formulation
Maxwell's equations can also be expressed in terms of potentials as follows:
Name | Gaussian units | SI units (Wb) | SI units (A⋅m) |
---|---|---|---|
Maxwell's equations (assuming Lorenz gauge )
|
|||
Lorenz gauge condition | |||
Relation to fields |
where
Tensor formulation
Maxwell's equations in the language of tensors makes Lorentz covariance clear. We introduce electromagnetic tensors and preliminary four-vectors in this article as follows:
Name | Notation | Gaussian units | SI units (Wb or A⋅m) |
---|---|---|---|
Electromagnetic tensor | |||
Dual electromagnetic tensor
|
|||
Four-current | |||
Four-potential | |||
Four-force |
where:
- The signature of the Minkowski metric is (+ − − −).
- The electromagnetic tensor and its Hodge dual are antisymmetric tensors:
The generalized equations are:[25][26]
Maxwell equations | Gaussian units | SI units (Wb) | SI units (A⋅m) |
---|---|---|---|
Ampère–Gauss law | |||
Faraday–Gauss law | |||
Lorentz force law |
Name | Gaussian units | SI units (Wb) | SI units (A⋅m) |
---|---|---|---|
Maxwell's equations | |||
Lorenz gauge condition | |||
Relation to fields (Cabibbo–Ferrari-Shanmugadhasan relation) |
|
|
where the εαβμν is the Levi-Civita symbol.
Duality transformation
The generalized Maxwell's equations possess a certain symmetry, called a duality transformation. One can choose any real angle ξ, and simultaneously change the fields and charges everywhere in the universe as follows (in Gaussian units):[29]
Charges and currents | Fields |
---|---|
where the primed quantities are the charges and fields before the transformation, and the unprimed quantities are after the transformation. The fields and charges after this transformation still obey the same Maxwell's equations.
Because of the duality transformation, one cannot uniquely decide whether a particle has an electric charge, a magnetic charge, or both, just by observing its behavior and comparing that to Maxwell's equations. For example, it is merely a convention, not a requirement of Maxwell's equations, that electrons have electric charge but not magnetic charge; after a ξ = π/2 transformation, it would be the other way around. The key empirical fact is that all particles ever observed have the same ratio of magnetic charge to electric charge.[29] Duality transformations can change the ratio to any arbitrary numerical value, but cannot change that all particles have the same ratio. Since this is the case, a duality transformation can be made that sets this ratio at zero, so that all particles have no magnetic charge. This choice underlies the "conventional" definitions of electricity and magnetism.[29]
Dirac's quantization
One of the defining advances in quantum theory was Paul Dirac's work on developing a relativistic quantum electromagnetism. Before his formulation, the presence of electric charge was simply "inserted" into the equations of quantum mechanics (QM), but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM.[30] That is to say, we can maintain the form of Maxwell's equations and still have magnetic charges.
Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationary magnetic monopole, which would not exert any forces on each other. Classically, the electromagnetic field surrounding them has a momentum density given by the Poynting vector, and it also has a total angular momentum, which is proportional to the product qeqm, and is independent of the distance between them.
Quantum mechanics dictates, however, that angular momentum is quantized as a multiple of ħ, so therefore the product qeqm must also be quantized. This means that if even a single magnetic monopole existed in the universe, and the form of
Although it would be possible simply to
However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field is proportional to the
Because the electron returns to the same point after the full trip around the equator, the phase φ of its wave function eiφ must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π. This is known as the Dirac quantization condition. In various units, this condition can be expressed as:
Units Condition SI units (ampere-meter convention) Gaussian-cgs units
where ε0 is the
The hypothetical existence of a magnetic monopole would imply that the electric charge must be quantized in certain units; also, the existence of the electric charges implies that the magnetic charges of the hypothetical magnetic monopoles, if they exist, must be quantized in units inversely proportional to the elementary electric charge.
At the time it was not clear if such a thing existed, or even had to. After all, another theory could come along that would explain charge quantization without need for the monopole. The concept remained something of a curiosity. However, in the time since the publication of this seminal work, no other widely accepted explanation of charge quantization has appeared. (The concept of local gauge invariance—see
If we maximally extend the definition of the vector potential for the southern hemisphere, it is defined everywhere except for a
The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline from spacetime); in more sophisticated theories, it is superseded by a smooth solution such as the 't Hooft–Polyakov monopole.
Topological interpretation
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
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:
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 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 e, solenoids with a flux of 2π/e have no interference fringes, because the phase factor for any charged particle is exp(2πi) = 1. Such a solenoid, if thin enough, is quantum-mechanically invisible. If such a solenoid were to carry a flux of 2π/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.
Grand unified theories
In a U(1) gauge group with quantized charge, the group is a circle of radius 2π/e. Such a U(1) gauge group is called
The case of the U(1) gauge group is a special case because all its
GUTs lead to compact U(1) gauge groups, so they explain charge quantization in a way that seems logically independent from magnetic monopoles. However, the explanation is essentially the same, because in any GUT that breaks down into a U(1) gauge group at long distances, there are magnetic monopoles.
The argument is topological:
- The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to group elements infinitesimally close to the identity.
- If you imagine a big sphere in space, you can deform an infinitesimal loop that starts and ends at the north pole as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts and ends at the north pole) then let it shrink back to a little loop while going over the eastern hemisphere. This is called lassoing the sphere.
- Lassoing is a sequence of loops, so the holonomy maps it to a sequence of group elements, a continuous path in the gauge group. Since the loop at the beginning of the lassoing is the same as the loop at the end, the path in the group is closed.
- If the group path associated to the lassoing procedure winds around the U(1), the sphere contains magnetic charge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere.
- Since the holonomy at the beginning and at the end is the identity, the total magnetic flux is quantized. The magnetic charge is proportional to the number of windings N, the magnetic flux through the sphere is equal to 2πN/e. This is the Dirac quantization condition, and it is a topological condition that demands that the long distance U(1) gauge field configurations be consistent.
- When the U(1) gauge group comes from breaking a contractible. Lie groups are homogeneous, so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the covering group ends at P, which is a lift of the identity. Going around the loop twice gets you to P2, three times to P3, all lifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. This number of times one has to traverse the loop to make it contractible is small, for example if the GUT group is SO(3), the covering group is SU(2), and going around any loop twice is enough.
- This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopole configuration to unwind itself at short distances, at the cost of not staying in the U(1). To do this with as little energy as possible, you should leave only the U(1) gauge group in the neighborhood of one point, which is called the core of the monopole. Outside the core, the monopole has only magnetic field energy.
Hence, the Dirac monopole is a topological defect in a compact U(1) gauge theory. When there is no GUT, the defect is a singularity – the core shrinks to a point. But when there is some sort of short-distance regulator on spacetime, the monopoles have a finite mass. Monopoles occur in lattice U(1), and there the core size is the lattice size. In general, they are expected to occur whenever there is a short-distance regulator.
String theory
In the universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be a black hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that the mass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, the lightest charged particles cannot be too heavy.[32] The lightest monopole should have a mass less than or comparable to its charge in natural units.
So in a consistent holographic theory, of which
Mathematical formulation
In mathematics, a (classical) gauge field is defined as a connection over a principal G-bundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately.
A connection on a G-bundle tells you how to glue fibers together at nearby points of M. It starts with a continuous symmetry group G that acts on the fiber F, and then it associates a group element with each infinitesimal path. Group multiplication along any path tells you how to move from one point on the bundle to another, by having the G element associated to a path act on the fiber F.
In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added on as an afterthought. In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of characteristic classes in algebraic topology is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Note that a connection over a trivial bundle can never give us a nontrivial principal bundle.
If spacetime is the space of all possible connections of the G-bundle is
A principal G-bundle over S2 is defined by covering S2 by two
So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not
The total magnetic flux is none other than the first
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to d + 1 dimensions with d ≥ 2 in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d − 3. Another way is to examine the type of topological singularity at a point with the homotopy group πd−2(G).
Grand unified theories
In more recent years, a new class of theories has also suggested the existence of magnetic monopoles.
During the early 1970s, the successes of
The majority of particles appearing in any quantum field theory are unstable, and they decay into other particles in a variety of reactions that must satisfy various
The dyons in these GUTs are also stable, but for an entirely different reason. The dyons are expected to exist as a side effect of the "freezing out" of the conditions of the early universe, or a symmetry breaking. In this scenario, the dyons arise due to the configuration of the vacuum in a particular area of the universe, according to the original Dirac theory. They remain stable not because of a conservation condition, but because there is no simpler topological state into which they can decay.
The length scale over which this special vacuum configuration exists is called the correlation length of the system. A correlation length cannot be larger than causality would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the metric of the expanding universe. According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place.
Cosmological models of the events following the
Many of the other particles predicted by these GUTs were beyond the abilities of current experiments to detect. For instance, a wide class of particles known as the X and Y bosons are predicted to mediate the coupling of the electroweak and strong forces, but these particles are extremely heavy and well beyond the capabilities of any reasonable particle accelerator to create.
Searches for magnetic monopoles
Experimental searches for magnetic monopoles can be placed in one of two categories: those that try to detect preexisting magnetic monopoles and those that try to create and detect new magnetic monopoles.
Passing a magnetic monopole through a coil of wire induces a net current in the coil. This is not the case for a magnetic dipole or higher order magnetic pole, for which the net induced current is zero, and hence the effect can be used as an unambiguous test for the presence of magnetic monopoles. In a wire with finite resistance, the induced current quickly dissipates its energy as heat, but in a
According to standard inflationary cosmology, magnetic monopoles produced before inflation would have been diluted to an extremely low density today. Magnetic monopoles may also have been produced thermally after inflation, during the period of reheating. However, the current bounds on the reheating temperature span 18 orders of magnitude and as a consequence the density of magnetic monopoles today is not well constrained by theory.
There have been many searches for preexisting magnetic monopoles. Although there has been one tantalizing event recorded, by Blas Cabrera Navarro on the night of February 14, 1982 (thus, sometimes referred to as the "Valentine's Day Monopole"[37]), there has never been reproducible evidence for the existence of magnetic monopoles.[13] The lack of such events places an upper limit on the number of monopoles of about one monopole per 1029 nucleons.
Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole in
.High-energy particle colliders have been used to try to create magnetic monopoles. Due to the conservation of magnetic charge, magnetic monopoles must be created in pairs, one north and one south. Due to conservation of energy, only magnetic monopoles with masses less than half of the center of mass energy of the colliding particles can be produced. Beyond this, very little is known theoretically about the creation of magnetic monopoles in high-energy particle collisions. This is due to their large magnetic charge, which invalidates all the usual calculational techniques. As a consequence, collider-based searches for magnetic monopoles cannot, as yet, provide lower bounds on the mass of magnetic monopoles. They can however provide upper bounds on the probability (or cross section) of pair production, as a function of energy.
The ATLAS experiment at the Large Hadron Collider currently has the most stringent cross section limits for magnetic monopoles of 1 and 2 Dirac charges, produced through Drell–Yan pair production. A team led by Wendy Taylor searches for these particles based on theories that define them as long lived (they do not quickly decay), as well as being highly ionizing (their interaction with matter is predominantly ionizing). In 2019 the search for magnetic monopoles in the ATLAS detector reported its first results from data collected from the LHC Run 2 collisions at center of mass energy of 13 TeV, which at 34.4 fb−1 is the largest dataset analyzed to date.[39]
The
The astrophysicist
"Monopoles" in condensed-matter systems
Since around 2003, various
A true magnetic monopole would be a new
The monopoles studied by condensed-matter groups have none of these properties. They are not a new elementary particle, but rather are an
There are a number of examples in
Some researchers use the term magnetricity to describe the manipulation of magnetic monopole quasiparticles in spin ice,[51][52][50][53] in analogy to the word "electricity".
One example of the work on magnetic monopole quasiparticles is a paper published in the journal Science in September 2009, in which researchers described the observation of quasiparticles resembling magnetic monopoles. A single crystal of the spin ice material dysprosium titanate was cooled to a temperature between 0.6 kelvin and 2.0 kelvin. Using observations of neutron scattering, the magnetic moments were shown to align into interwoven tubelike bundles resembling Dirac strings. At the defect formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were able to control the density and orientation of these strings. A contribution to the heat capacity of the system from an effective gas of these quasiparticles was also described.[16][54] This research went on to win the 2012 Europhysics Prize for condensed matter physics.
In another example, a paper in the February 11, 2011 issue of Nature Physics describes creation and measurement of long-lived magnetic monopole quasiparticle currents in spin ice. By applying a magnetic-field pulse to crystal of dysprosium titanate at 0.36 K, the authors created a relaxing magnetic current that lasted for several minutes. They measured the current by means of the electromotive force it induced in a solenoid coupled to a sensitive amplifier, and quantitatively described it using a chemical kinetic model of point-like charges obeying the Onsager–Wien mechanism of carrier dissociation and recombination. They thus derived the microscopic parameters of monopole motion in spin ice and identified the distinct roles of free and bound magnetic charges.[53]
In
Updates to the theoretical and experimental searches in matter can be found in the reports by G. Giacomelli (2000) and by S. Balestra (2011) in the Bibliography section.
See also
Notes
- arXiv:physics/0508099v1, eqn (4), for example.
References
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- ^ "Particle Data Group summary of magnetic monopole search" (PDF). lbl.gov.
- ^ Wen, Xiao-Gang; Witten, Edward, "Electric and magnetic charges in superstring models", Nuclear Physics B, Volume 261, pp. 651–677
- ^ S. Coleman, "The Magnetic Monopole 50 years Later", reprinted in Aspects of Symmetry
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- ^ Pierre Curie (1894). "Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre" [On the possible existence of magnetic conductivity and free magnetism]. Séances de la Société Française de Physique (in French). Paris: 76–77.
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"Magnetic Monopoles Detected in a Real Magnet for the First Time". Science Daily. September 4, 2009. Retrieved September 4, 2009.
- ^ Symmetry Breaking, January 29, 2009. Retrieved January 31, 2009.
- ^ a b "Magnetic monopoles spotted in spin ices", Physics World, September 3, 2009. "Oleg Tchernyshyov at Johns Hopkins University [a researcher in this field] cautions that the theory and experiments are specific to spin ices, and are not likely to shed light on magnetic monopoles as predicted by Dirac."
- ^ S2CID 124109501.
This is not the first time that physicists have created monopole analogues. In 2009, physicists observed magnetic monopoles in a crystalline material called spin ice, which, when cooled to near-absolute zero, seems to fill with atom-sized, classical monopoles. These are magnetic in a true sense, but cannot be studied individually. Similar analogues have also been seen in other materials, such as in superfluid helium. ... Steven Bramwell, a physicist at University College London who pioneered work on monopoles in spin ices, says that the [2014 experiment led by David Hall] is impressive, but that what it observed is not a Dirac monopole in the way many people might understand it. 'There's a mathematical analogy here, a neat and beautiful one. But they're not magnetic monopoles.'
- ISBN 978-0-321-85656-2.
- ISBN 978-0-07-051400-3.
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- ^ Jackson 1999, section 6.11, equation (6.153), p. 275
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- ^ Alvarez, Luis W. "Analysis of a Reported Magnetic Monopole". In Kirk, W. T. (ed.). Proceedings of the 1975 international symposium on lepton and photon interactions at high energies. International symposium on lepton and photon interactions at high energies, Aug 21, 1975. p. 967. Archived from the original on February 4, 2009. Retrieved May 25, 2008.
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- All About Space. No. 24. April 2014.]
If the structures of the magnetic fields appear to be magnetic monopoles, that are macroscopic in size, then this is a wormhole.
[author missing - ^ "Quantised Singularities in the Electromagnetic Field" Paul Dirac, Proceedings of the Royal Society, May 29, 1931. Retrieved February 1, 2014.
- Particle data group, updated August 2015 by D. Milstead and E.J. Weinberg. "To date there have been no confirmed observations of exotic particles possessing magnetic charge."
- .
Magnetic monopoles have also inspired condensed-matter physicists to discover analogous states and excitations in systems such as spin ices and Bose–Einstein condensates. However, despite the importance of those developments in their own fields, they do not resolve the question of the existence of real magnetic monopoles. Therefore, the search continues.
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D.J.P. Morris; D.A. Tennant; S.A. Grigera; B. Klemke; C. Castelnovo; R. Moessner; C. Czter-nasty; M. Meissner; K.C. Rule; J.-U. Hoffmann; K. Kiefer; S. Gerischer; D. Slobinsky & R.S. Perry (September 3, 2009) [2009-07-09]. "Dirac Strings and Magnetic Monopoles in Spin Ice Dy2Ti2O7". S2CID 206522398.
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- Balestra, S. (2011), Magnetic Monopole Bibliography-II, arXiv:1105.5587
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- Lacava, F. (2022). Classical Electrodynamics: From Image Charges to the Photon Mass and Magnetic Monopoles (2nd ed.). Springer. ISBN 978-3-031-05098-5.
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External links
This article incorporates material from N. Hitchin (2001) [1994], "Magnetic Monopole",